Critical points (multivariable)

Critical points in multivariable calculus extend the concept from single-variable calculus to functions of multiple variables. For single-variable critical points, see [[ ../differential_calculus/Critical Points ]].

Exercises

  1. Consider the function $f(x,y) = y^2 - x^2 + x^2y - xy + 3y$ a) Compute the gradient. b) Find all the critical points.

  2. Consider the function $f(x,y) = x^3 + x - 4xy - 2y^2$ a) Compute the gradient. b) Find all the critical points.

  3. Consider the function $f(x,y) = \frac{1}{2}x^2 + 7xy + 2y^2 + 4x + 8y$ a) Determine the gradient. b) Find all the critical points. c) Classify the critical points.

  4. Consider the function $f(x,y) = x^3y^2+y^2-xy+x$ a) Determine the gradient. b) Find all the critical points.

  5. Consider the function $f(x,y) = xy(x-1)$ a) Determine the gradient. b) Find all the critical points.

  6. Consider the function $f(x,y) = \frac{x^3y}{3} + \frac{x^2y}{2} + \frac{1}{2}y^2$ a) Determine the gradient. b) Find all the critical points. c) Establish if $A = (\frac{-3}{2}, 0)$ and $B = (-1, \frac{-1}{6})$ are maximum, minimum or saddle points.

  7. Consider the function $f(x,y) = e^x(2x^2-xy+y^2)$ a) Establish if $A = (0,0)$ is a maximum, minimum or a saddle point.

  8. Find the local extrema of the following functions: a) $f(x, y) = x^2+y^2$ b) $f(x, y) = x^2-xy+y^2+3x$ c) $f(x, y) = x^2+xy+y^2-6x+6$

  9. Classify all the critical points of the function $f(x, y) = x^2 - \cos y$.