Find the absolute maximum and absolute minimum of the function f(x, y)=2x3+y4f(x, y)=2x3+y4 on the region {(x, y)|x2+y2≤64}{(x, y)|x2+y2≤64} ignore unneeded answer blanks, and list points lexicographically.


The answer maybbe c

8 2/3 is the answer

step-by-step explanation: just use a calculator

Check where the first-order partial derivatives vanish to find any critical points within the given region:


The Hessian for this function is

\mathbf H(x,y)=\begin{bmatrix}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{bmatrix}=\begin{bmatrix}12x&0\\0&12y^2\end{bmatrix}

with \det\mathbf H(0,0)=0, so unfortunately the second partial derivative test fails. However, if we take x=0 we see that f(x,y)0 for different values of y; if we take y=0 we see f(x,y) takes on both positive and negative values. This indicates (0, 0) is neither the site of an extremum nor a saddle point.

Now check for points along the boundary. We can parameterize the boundary by

(x,y)=(8\cos t,8\sin t)

with 0\le t\le2\pi. This turns f(x,y) into a univariate function F(t):

F(t)=f(8\cos t,8\sin t)=2^{10}\cos^3t+2^{12}\sin^4t

\implies F'(t)=3\cdot2^{10}\cos^2t(-\sin t)+2^{14}\sin^3t\cos t=2^{10}\sin t\cos t(16\sin^2t-3\cos t)

F'(t)=0\implies\begin{matrix}\sin t=0\implies t=0,\,t=\pi\\\\\cos t=0\implies t=\dfrac\pi2,\,t=\dfrac{3\pi}2\\\\16\sin^2t-3\cos t=0\implies t=2\tan^{-1}\sqrt{\dfrac{\sqrt{1033}-32}3},\,t=2\pi-2\tan^{-1}\sqrt{\dfrac{\sqrt{1033}-32}3}\end{matrix}

At these critical points, we get







We only care about 3 of these results.




So to recap, we found that f(x,y) attains

a maximum value of 4096 at the points (0, 8) and (0, -8), anda minimum value of -1024 at the point (-8, 0).

Do you know the answer?

Other questions on the subject: Mathematics

Option AStep-by-step explanation:Given that a square stained glass window is divided into four congruent triangular sections by iron edging to represent the seasons of the yearEach...Read More
3 more answers
where are the images of the functions i can only see the question you typed.step-by-step explanation:...Read More
1 more answers
answer: sherkstep-by-step explanation: just give up and watch the best anime ever sherk...Read More
1 more answers
Mathematics, 22.06.2019, nadine3782
all group life insurance is term insurance. actually it is annual renewable term and it is rated on the average age and claims experience of the entire group, which is called "expe...Read More
3 more answers
Mathematics, 22.06.2019, madelyngv97
a)Annual Repayment Installment:$2,812.1b)Amount to be applied to interest:$1040Amount to be applied to principal:  $1772.10Explanation:The annual payment which will be used to offs...Read More
1 more answers