Caitlin Esparza

2022-02-12

Find the critical points of the following functions. Use the Second Derivative Test to determine (if possible) whether each critical point corresponds to a local maximum, local minimum, or saddle point. Confirm your results using a graphing utility.

zerogirlg16

Step 1
Differentiate $f\left(x,y\right)={\left(4x-1\right)}^{2}+{\left(2y+4\right)}^{2}+1$ partially with respect to x and y to obtain the values of ${f}_{x}$ and ${f}_{y}$.
$f\left(x,y\right)={\left(4x-1\right)}^{2}+{\left(2y+4\right)}^{2}+1$
${f}_{x}=\frac{\partial {\left(4x-1\right)}^{2}+{\left(2y+4\right)}^{2}+1}{\partial x}$
$=32x-8$
${f}_{y}=\frac{\partial {\left(4x-1\right)}^{2}+{\left(2y+4\right)}^{2}+1}{\partial y}$
$=8\left(y+2\right)$
Step 2
Equate ${f}_{x}$ and ${f}_{y}$ to 0 and solve for x and y.
${f}_{x}=0$
$32x-8=0$
$x=\frac{8}{32}$
$=\frac{1}{4}$
${f}_{y}=8\left(y+2\right)$
$0=8\left(y+2\right)$
$y=-2$
The critical point is .
Step 3
Differentiate ${f}_{x}$ and ${f}_{y}$ partially with the respect to x and y respectively to obtain the value of ${f}_{xx}$ and ${f}_{yy}$ and differentiate ${f}_{x}$ with respect to y to obtain the value of ${f}_{xy}$ equate ${f}_{xx}$ and ${f}_{yy}$ to A and C respectively and ${f}_{xy}$ to B.
${f}_{xx}=\frac{\partial \left(32x-8\right)}{\partial x}$
$=32$
$A=32$
${f}_{yy}=\frac{\partial \left(8\left(y+2\right)\right)}{\partial y}$
$=8$
$C=8$
${f}_{xy}=\frac{\partial \left(32x-8\right)}{\partial y}$
$=0$
$B=0$
Step 4
Substitute the values of A,B,C in $AC-{B}^{2}$ and obtain the value of the expression.
$AC-{B}^{2}=\left(32\right)\left(8\right)-{0}^{2}$

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