Question

Give an example of a continuously differentiable function from
R2
to $\mathbb{R}$ , which has an isolated local maximum at (0,0) and in (-17,9) and (0,3) an isolated local minimum in each case. Justify your answer.

Answer 1

A function having local maxima at (0,0) and local minimum at (-17,9) is

2 f(2,y) = 1- + 172 92

which possesses local maximum value equal to at (0,0) and local minimum value at (-17,9) in the subset

$D=\{(x,y)\in\mathbb R^2: \frac{x^2}{17^2}+\frac{y^2}{9^2}\le 1\}$

Similarly, for local maximum at (0,0) and local minimum at (0,3) an example is

$g(x,y)=\left [ 1-\left (x^2+\frac{y^2}{3^2}\right )\right ]^2$

which attains local maximum value equal to at (0,0) and local minimum value equal to at (0,3) in the subset

$E=\{(x,y)\in\mathbb R^2:x^2+\frac{y^2}{3^2}\le 1\}$