Chapter 12

Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function. \(f(x, y)=1-2 x^{2}-3 y^{2}\)

Let \(f(x, y)=x^{2}+2 y^{2}\). a. Find \(f_{x}(2,1)\) and \(f_{y}(2,1)\). b. Interpret the numbers in part (a) as slopes. c. Interpret the numbers in part (a) as rates of change.

Let \(f(x, y)=2 x+3 y-4\). Compute \(f(0,0), f(1,0), f(0,1)\), \(f(1,2)\), and \(f(2,-1) .\)

Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function. \(f(x, y)=x^{2}-x y+y^{2}+1\)

Let \(f(x, y)=9-x^{2}+x y-2 y^{2}\). a. Find \(f_{x}(1,2)\) and \(f_{y}(1,2)\). b. Interpret the numbers in part (a) as slopes. c. Interpret the numbers in part (a) as rates of change.

Let \(g(x, y)=2 x^{2}-y^{2}\). Compute \(g(1,2), g(2,1), g(1,1)\), \(g(-1,1)\), and \(g(2,-1) .\)

Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function. \(f(x, y)=x^{2}-y^{2}-2 x+4 y+1\)

Find the first partial derivatives of the function. \(f(x, y)=2 x+3 y+5\)

Let \(f(x, y)=x^{2}+2 x y-x+3\). Compute \(f(1,2), f(2,1)\), \(f(-1,2)\), and \(f(2,-1)\).

Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function. \(f(x, y)=2 x^{2}+y^{2}-4 x+6 y+3\)

Find the first partial derivatives of the function. \(f(x, y)=2 x y\)

Let \(h(x, y)=(x+y) /(x-y) .\) Compute \(h(0,1), h(-1,1)\), \(h(2,1)\), and \(h(\pi,-\pi) .\)

Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function. \(f(x, y)=x^{2}+2 x y+2 y^{2}-4 x+8 y-1\)

Find the first partial derivatives of the function. \(g(x, y)=2 x^{2}+4 y+1\)

Let \(g(s, t)=3 s \sqrt{t}+t \sqrt{s}+2\). Compute \(g(1,2), g(2,1)\), \(g(0,4)\), and \(g(4,9)\)

Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function. \(f(x, y)=x^{2}-4 x y+2 y^{2}+4 x+8 y-1\)

Find the first partial derivatives of the function. \(f(x, y)=1+x^{2}+y^{2}\)

Let \(f(x, y)=x y e^{x^{2}+y^{2}}\). Compute \(f(0,0), f(0,1), f(1,1)\), and \(f(-1,-1)\)

Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function. \(f(x, y)=2 x^{3}+y^{2}-9 x^{2}-4 y+12 x-2\)

Find the first partial derivatives of the function. \(f(x, y)=\frac{2 y}{x^{2}}\)

Let \(h(s, t)=s \ln t-t \ln s\). Compute \(h(1, e), h(e, 1)\), and \(h(e, e)\)

Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function. \(f(x, y)=2 x^{3}+y^{2}-6 x^{2}-4 y+12 x-2\)

Find the first partial derivatives of the function. \(f(x, y)=\frac{x}{1+y}\)

Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function. \(f(x, y)=x^{3}+y^{2}-2 x y+7 x-8 y+4\)

Find the first partial derivatives of the function. \(g(u, v)=\frac{u-v}{u+v}\)

Let \(g(r, s, t)=r e^{s / t} .\) Compute \(g(1,1,1), g(1,0,1)\), and \(g(-1,-1,-1)\)

Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function. \(f(x, y)=2 y^{3}-3 y^{2}-12 y+2 x^{2}-6 x+2\)

Find the first partial derivatives of the function. \(f(x, y)=\frac{x^{2}-y^{2}}{x^{2}+y^{2}}\)

Let \(g(u, v, w)=\left(u e^{v w}+v e^{w w}+w e^{u t}\right) /\left(u^{2}+v^{2}+w^{2}\right)\). Compute \(g(1,2,3)\) and \(g(3,2,1)\).

Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function. \(f(x, y)=x^{3}-3 x y+y^{3}-2\)

Find the first partial derivatives of the function. \(f(s, t)=\left(s^{2}-s t+t^{2}\right)^{3}\)

Sketch the level curves of the function corresponding to each value of \(z\). . \(f(x, y)=2 x+3 y\)

Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function. \(f(x, y)=x^{3}-2 x y+y^{2}+5\)

Find the first partial derivatives of the function. \(g(s, t)=s^{2} t+s t^{-3}\)

Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function. \(f(x, y)=x y+\frac{4}{x}+\frac{2}{y}\)

Find the first partial derivatives of the function. \(f(x, y)=\left(x^{2}+y^{2}\right)^{2 / 3}\)

Find the first partial derivatives of the function. \(f(x, y)=x \sqrt{1+y^{2}}\)

Sketch the level curves of the function corresponding to each value of \(z\). \(f(s, t)=\sqrt{s^{2}+t^{2}}\)

Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function. \(f(x, y)=x^{2}-e^{y^{2}}\)

Find the first partial derivatives of the function. \(f(x, y)=e^{x y+1}\)

Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function. \(f(x, y)=e^{x^{2}-y^{2}}\)

Find the first partial derivatives of the function. \(f(x, y)=\left(e^{x}+e^{y}\right)^{5}\)

Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function. \(f(x, y)=e^{x^{2}+y^{2}}\)

Find the first partial derivatives of the function. \(f(x, y)=x \ln y+y \ln x\)

Find the first partial derivatives of the function. \(f(x, y)=x^{2} e^{y^{2}}\)

Sketch the level curves of the function corresponding to each value of \(z\). \(h(u, v)=\sqrt{4-u^{2}-v^{2}}\)

Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function. \(f(x, y)=\ln \left(1+x^{2}+y^{2}\right)\)

Sketch the level curves of the function corresponding to each value of \(z\). \(f(x, y)=2 x+3 y ; z=-2,-1,0,1,2\)

Find the critical point(s) of the function. Then use the second derivative test to classify the nature of each point, if possible. Finally, determine the relative extrema of the function. \(f(x, y)=x y+\ln x+2 y^{2}\)

Find the first partial derivatives of the function. \(f(x, y)=\frac{e^{x y}}{x+y}\)