The critical points of h(x,y) occur wherever its partial derivatives [tex]h_x[/tex] and [tex]h_y[/tex] vanish simultaneously. We have
[tex]h_x = 8-4y-8x = 0 \implies y=2-2x \\\\ h_y = 10-4x-12y^2 = 0 \implies 2x+6y^2=5[/tex]
Substitute y in the second equation and solve for x, then for y :
[tex]2x+6(2-2x)^2=5 \\\\ 24x^2-46x+19=0 \\\\ \implies x=\dfrac{23\pm\sqrt{73}}{24}\text{ and }y=\dfrac{1\mp\sqrt{73}}{12}[/tex]
This is to say there are two critical points,
[tex](x,y)=\left(\dfrac{23+\sqrt{73}}{24},\dfrac{1-\sqrt{73}}{12}\right)\text{ and }(x,y)=\left(\dfrac{23-\sqrt{73}}{24},\dfrac{1+\sqrt{73}}{12}\right)[/tex]
To classify these critical points, we carry out the second partial derivative test. h(x,y) has Hessian
[tex]H(x,y) = \begin{bmatrix}h_{xx}&h_{xy}\\h_{yx}&h_{yy}\end{bmatrix} = \begin{bmatrix}-8&-4\\-4&-24y\end{bmatrix}[/tex]
whose determinant is [tex]192y-16[/tex]. Now,
• if the Hessian determinant is negative at a given critical point, then you have a saddle point
• if both the determinant and [tex]h_{xx}[/tex] are positive at the point, then it's a local minimum
• if the determinant is positive and [tex]h_{xx}[/tex] is negative, then it's a local maximum
• otherwise the test fails
We have
[tex]\det\left(H\left(\dfrac{23+\sqrt{73}}{24},\dfrac{1-\sqrt{73}}{12}\right)\right) = -16\sqrt{73} < 0[/tex]
while
[tex]\det\left(H\left(\dfrac{23-\sqrt{73}}{24},\dfrac{1+\sqrt{73}}{12}\right)\right) = 16\sqrt{73}>0 \\\\ \text{ and } \\\\ h_{xx}\left(\dfrac{23+\sqrt{73}}{24},\dfrac{1-\sqrt{73}}{12}\right)=-8 < 0[/tex]
So, we end up with
[tex]h\left(\dfrac{23+\sqrt{73}}{24},\dfrac{1-\sqrt{73}}{12}\right)=-\dfrac{4247+37\sqrt{73}}{72} \text{ (saddle point)}\\\\\text{ and }\\\\h\left(\dfrac{23-\sqrt{73}}{24},\dfrac{1+\sqrt{73}}{12}\right)=-\dfrac{4247-37\sqrt{73}}{72} \text{ (local max)}[/tex]
simplify
-12 divided by 3 x (-8 + (-4) ^2 - 6) + 2
[tex] \large \mathfrak{Solution : }[/tex]
-12 ÷ 3 × (-8 + (-4)² - 6) + 2-12 ÷ 3 × (-8 + 16 - 6) + 2-12 ÷ 3 × 2 + 2 -4 × 2 + 2-8 + 2-6simplify 5 - (3 a + 1) + 2 a
Step-by-step explanation:
5-3a-1+2a
5-1-3a+2a
4-1a
I know the answer don’t know how though
Answer:
16384Step-by-step explanation:
As,
[tex] {2}^{15} [/tex]
Meaning that 2 multiplied 15 times here,
Equals 32 768
Now,
[tex] {2}^{14} [/tex]
Meaning that 2 multiplied 15 times then divided by one time 2,
Equals 32 768 ÷ 2 = 16384 (Ans)
pls help im a little bit sтuрid
Step-by-step explanation:
a⁴+16a²b-2ab-a²+64b²-b²we can also write them as,
a⁴+16a²b+64b²-a²-2ab-b²Now, from -a²-2ab-b² if we take - as common term then we get,
a⁴+16a²b+64b²-(a²+2ab+b²)Now, by using the concept of a²+2ab+b²=(a+b)²
(a²)²+2×a²×8b+(8b)²-{(a)²+2×a×b+(b)²}(a²+8b)²-(a+b)²I hope it helped U
stay safe stay happy
Pls I am really struggling here
How do you know the end behavior of a polynomial function if the first number is a variable? Do you just move on to the next term that is a number
Answer:
You need to bring the function to the standard form.
The term with highest degree exponent is the leading term and its coefficient is the leading coefficient.
Let it be axⁿ.
Depending on the n and a, the end behavior of the function will change.
Case 1a > 0, n - is oddThis is an odd function and:
x → -∞ ⇒ f(x) → -∞x → ∞ ⇒ f(x) → ∞Case 2a < 0, n - is oddThis is an odd function and:
x → -∞ ⇒ f(x) → ∞x → ∞ ⇒ f(x) → -∞Case 3a > 0, n - is evenThis is an even function and:
x → -∞ ⇒ f(x) ⇒ ∞x → ∞ ⇒ f(x) ⇒ ∞Case 4a < 0, n - is evenThis is an even function and:
x → - ∞ ⇒ f(x) ⇒ - ∞x → ∞ ⇒ f(x) ⇒ - ∞Answer:
Yeah your right
Step-by-step explanation:
The odometer on your car read 53,305 at this time last year. It now reads 79,964. Estimate the number of miles your car has been driven during the past year. (Estimate to the nearest thousand miles.)
Answer:
26659
Step-by-step explanation:
subtracting the recent meter and the last time meter
(6x-5y+4)dy+(y-2x-1)dx=0
(6x - 5y + 4) dy + (y - 2x - 1) dx = 0
(6x - 5y + 4) dy = (2x - y + 1) dx
dy/dx = (2x - y + 1) / (6x - 5y + 4)
Let X = x - a and Y = y - b. We want to find constants a and b such that
dY/dX = (a rational function)
where the numerator and denominator on the right side are free of constant terms. Substituting x and y in the equation, we have
dY/dX = (2 (X + a) - (Y + b) + 1) / (6 (X + a) - 5 (Y + b) + 4)
dY/dX = (2X - Y + 2a - b + 1) / (6X - 5Y + 6a - 5b + 4)
Then we solve for a and b in the system,
2a - b + 1 = 0
6a - 5b + 4 = 0
==> a = -1/4 and b = 1/2
With these constants, the equation reduces to
dY/dX = (2X - Y) / (6X - 5Y)
Now substitute Y = VX and dY/dX = X dV/dX + V :
X dV/dX + V = (2X - VX) / (6X - 5VX)
The equation becomes separable after some simplification:
X dV/dX + V = (2 - V) / (6 - 5V)
X dV/dX = (2 - V) / (6 - 5V) - V
X dV/dX = (2 - V - (6 - 5V)) / (6 - 5V)
X dV/dX = (4V - 4) / (6 - 5V)
- (5V - 6) / (4V - 4) dV = 1/X dX
Integrate both sides:
-5/4 V + 1/4 ln|4V - 4| = ln|X| + C
Extract a constant from the logarithm on the left:
-5/4 V + 1/4 (ln(4) + ln|V - 1|) = ln|X| + C
-5/4 V + 1/4 ln|V - 1| = ln|X| + C
-5V + ln|V - 1| = 4 ln|X| + C
Get this back in terms of Y :
-5Y/X + ln|Y/X - 1| = 4 ln|X| + C
Now get the solution in terms of y and x :
-5 (y - 1/2)/(x + 1/4) + ln|(y - 1/2)/(x + 1/4) - 1| = 4 ln|x + 1/4| + C
With some manipulation of constants and logarithms, and a bit of algebra, we can rewrite this solution as
-5 (4y - 2)/(4x + 1) + ln|(4y - 4x - 3)/(4x + 1)| = 4 ln|x + 1/4| + 4 ln(4) + C
-5 (4y - 2)/(4x + 1) + ln|(4y - 4x - 3)/(4x + 1)| = 4 ln|4x + 1| + C
-5 (4y - 2)/(4x + 1) + ln|4y - 4x - 3| - ln|4x + 1| = 4 ln|4x + 1| + C
-5 (4y - 2)/(4x + 1) + ln|4y - 4x - 3| = 5 ln|4x + 1| + C
Find the midpoint of the line segment with end coordinates of: ( 2 , 0 ) and ( 8 , 8 )
Answer:
mid point is (1, 8)
Step-by-step explanation:
[tex]{ \bf{midpoint = ( \frac{x_{1} + x _{2} }{2} , \: \frac{y _{1} + y _{2} }{2} ) }}[/tex]
x1 is 2
x2 is 0
y1 is 8
y2 is 8
[tex] = ( \frac{2 + 0}{2} , \: \frac{8 + 8}{2} ) \\ \\ = ( \frac{2}{2} , \: \frac{16}{2} ) \\ \\ = (1, \: 8)[/tex]
A farmer performs a cross between two species of flowers and expects a distribution of 25% red, 50% pink, and 25% white flowers. Out of 224 flowers, the following numbers of red, pink, and white flowers were observed. Calculate the expected number of each type of flower.
Answer:
the number of pink would be 112 because 224 divided by 2 is 112 then the red and white are 56 cuz 112 divided by 2 is 56
Step-by-step explanation:
2. Consider the division operations below and choose whether the result can be rational,
irrational, or both.
Expressions
Rational
Irrational
O
O
Flá na por laskas
5
16
o
o
Answer:
All are rational.
Step-by-step explanation:
1)
=1/3/3/5
=1/3×5/3
=5/9
5/9 is non-terminating and recurring number so it is rational.
2)
4/3√9/5
=4/3*3/5
=4/5
4/5 is a terminating number it is rational.
3)
2*√16/√4
=2*4/2
=4
4 is a natural/whole number it is also rational.
Note:if you need to ask any questions please let me know.
a cake is sliced into 12 equal pieces 1/2 of the cake is: slices
Answer:
6
Step-by-step explanation:
1/2 of 12
1/2×12
6
i hope this will help
I need some help I’ve been trying to figure this out for a while now and I keep coming up with different answers. Match the pairs of equivalent expressions.
Answer:
dccxxydffffyyyrtttttttyhg
Jaxson did the following problem incorrectly. Choose the option that correctly identifies what Jaxson did
wrong.
Jaxson's work:
8 3/5 + 7 1/8 = 15 4/13
Answer:
4/13 very identifies what Jaxson did wrong.
Step-by-step explanation:
The addition of the whole number part of the fraction was well done that is (8+7)
What Jaxson did wrong in his work was the addition of the fractions (⅗+⅛). They have different denominators and thus can't be added straightforwardly. the denominator has to be a common value.
1 tenth + 2 tenths = blank tenths =
Answer:
3 tenths +4 tenths: 7 tenths
Step-by-step explanation:
2+1 :3
so
3+4:7
PLS HELP!! WILL AMRK BRAINLIEST IF RIGHT
9514 1404 393
Answer:
3 feet
Step-by-step explanation:
The equation with all the parameters filled in is ...
f(t) = -16t² +8t +2
The maximum height occurs at the vertex of the function, which can be written in vertex form as ...
f(t) = -16(t² -1/2t) +2
f(t) = -16(t² -1/2t +1/16) +2 +16(1/16)
f(t) = -16(x -1/4)² +3
The vertex of the graph is (1/4, 3), so the maximum height is 3 feet.
How many solutions are there to the equation below?
5x + 48 + 7 x = 12(x + 4)
A. 1
B. Infinitely many
C. 0
COM
Which expression is equivalent to the given expression? 2x^2-14 +24
Answer:
Equivalent expression is (x - 4)(x - 3)
Step-by-step explanation:
[tex] {2x}^{2} - 14x + 24[/tex]
find the factors:
[tex] = (x - 4)(x - 3)[/tex]
(2x-12)(x-2)
after multiplying you'll get the same
Peter mixes 4 1/2 cups of orange juice, 1 1/3 cups of ginger ale, 6 1/3 and oſ cups of strawberry lemonade to make some punch. What is the
total number of cups of punch that Peter makes?
If f(x) = x³ - 2, find f(3).
Answer:
25
Step-by-step explanation:
First you have to understand what f(3) means.
f(x) = x^3 - 2
f(3) means that wherever you see and x, you put in a 3.
f(3) = 3^3 - 2
f(3) = 3*3*3 - 2
f(3) = 27 - 2
f(3) = 25
Class 8 squares and square roots without adding find the sum of a) 1+3+5+7+9+11+13+15+17.
Answer:
given
n=9
a=1
d=3-1=2
Step-by-step explanation:
now , by formula,
sn = n/2(2a+(n-1)d)
= 9/2(2×1+(9-1)×2)
=9/2×18
=81
What are the solutions of the equation? X^2-16/25=0
x= 4/25 and x =-4/25
x = 8/25and x =-8/25
x = 16/25and x = -16/25
X =4/5 and x=-4/5
Answer:
x=4/5 ,x=-4/5
Step-by-step explanation:
Equation:
x²-16/25=0
x²=16/25
x=+-√16/25
x=+-4/5
Note:if you need to ask any question please let me know.
What’s equivalent to this
Answer:
a^(m-n)
Step-by-step explanation:
We have
a^m / a^n
Exponent rules state that
a^m / a^n = a^(m-n)
The mean number of guests on 5 nights is 90. When an extra night of guests is added the mean is 91 . What was the number of guests on the extra night ?
Answer:
96
Step-by-step explanation:
The mean number of 5 nights is 90. It means that the quantity of guests who came these nights is 90*5=450
The mean number of the six nights is 91. So the quantity of guests is 6*91=546
546-450= 96- the quantity of guests that came on the extra night
The sum of two number is -1/3, if one of the number is -12/3 ,find the other number? please help
Answer:
11/3
Step-by-step explanation:
-12/3 = -4
-4 + x = -1/3
X = -1/3 + 4
X = (-1+12)/3 = 11/3
11/3 + -12/3 = -1/3
Write the equation of the line parallel to y = 1/3x- 6 that passes through the point (-12,3) in slope-intercept form
Answer:
[tex]y=\frac{1}{3} x+7.[/tex]
Step-by-step explanation:
all the detail are in the attached picture, here is the short explanation step-by-step:
1) to select the values of slope and interception from the given equation;
2) using the rule of parallel lines to find the slope of the required line;
3) to write the required equation (the value of slope is known, the value of interception is inknown);
4) to calculate the unknown value of interception of the required equation after the substitution the coordinates of the point in the condition;
5) to make up the required equation using calculated slope and interception.
2 A car transporter can carry 13 cars. How many transporters are needed for 340 cars?
Answer:
27 car transporter
Step-by-step explanation:
Number of cars a car transporter can carry= 13
Number of cars there=340
number of transporters needed= 340÷13
= 27
Solve 6(x - 3) = 11x +7
Answer:
x = -5Step-by-step explanation:
Given,
6(x-3) = 11x + 7
=> 6x - 18 = 11x + 7
=> 11x - 6x = -18 - 7
=> 5x = -25
[tex] = > x = \frac{ - 25}{5} [/tex]
=> x = -5 (Ans)
[tex]\large\bf{\red{ \bullet}} \: \tt \:\: 6(x - 3) \: = \: 11x + 7[/tex]
[tex]\large\bf{\red{ \bullet}} \: \tt \:\:6x \: - \: 18 \: = \: 11x \: + \: 7[/tex]
[tex]\large\bf{\red{ \bullet}} \: \tt \:\: - 7 \: \: - 18 \: = \: 11x \: - \: 6x[/tex]
[tex]\large\bf{\red{ \bullet}} \: \tt \: \: - 25 \: = \: 5x[/tex]
[tex]\large\bf{\red{ \bullet}} \: \tt \: \cancel\frac{-25}{5} \: = \: x[/tex]
[tex]\large\bf{\red{ \bullet}} \: \tt \: - 5 \: = \: x[/tex]
[tex]\large\bf{\red{ \bullet}} \: \tt \: \: x \: = \: - 5[/tex]
can anyone plz with steps (not for useless man )
[tex]\boxed{\sf \displaystyle{\lim_{x\to a}}(x^n)=a^n}[/tex]
[tex]\\ \sf\longmapsto \displaystyle{\lim_{x\to \infty}}(\sqrt{3x}-\sqrt{x-5})[/tex]
[tex]\\ \sf\longmapsto \displaystyle{\lim_{x\to \infty}}\sqrt{3x}-\displaystyle{\lim_{x\to \infty}}\sqrt{x-5}[/tex]
[tex]\\ \sf\longmapsto \infty^{\dfrac{1}{2}}-\sqrt{\infty-5}[/tex]
[tex]\\ \sf\longmapsto \infty-\sqrt{\infty}[/tex]
[tex]\\ \sf\longmapsto\infty-\infty[/tex]
[tex]\\ \sf\longmapsto\infty[/tex]
How do you determine if the numbers are positive or negative?
Answer:
Positive integers have values greater than zero.
Negative integers have values less than zero.
Zero is neither positive nor negative.
Step-by-step explanation:
Answer: if they have got - before them or if the say negative
Step-by-step explanation:
Fill in the blank to make the number divisible by 3: 6178_9
Answer:
2, 5, or 8
Step-by-step explanation:
ccalculator go brr
Answer:
617829/3=205,943
617859/3=205,953
617889/3=205,963
Step-by-step explanation:
617819/3=205,939.666
617829/3=205,943
617839/3=205,946.333
617849/3=205,949.666
617859/3=205,953
617869/3=205,956.333
617879/3=205,959.666
617889/3=205,963
617899/3=205,966.333
