Bounded Homework Derivative Answers

[Ted123]

Homework Statement

Let $$f:[-1,1] \times \mathbb{R} \to\mathbb{R}$$ be a function.

If $f$ is defined by:

(i) $f(x,y) = 3\exp(x-y^2)$

then is the derivative with respect to $y$ bounded?

If $f$ is defined by:

(ii) $f(x,y) = 7\exp(y^2-x)$

then is the derivative with respect to $y$ bounded?

The Attempt at a Solution

For (i):

$\frac{\partial f}{\partial y} = -6y\exp(x-y^2) = -6y\exp(-y^2)\exp(x)$

and since $\exp(x) \leq e$ in the specified domain and since $y\exp(-y^2)$ is a bounded function on $\mathbb{R}$, $\frac{\partial f}{\partial y}$ is bounded - right?

For (ii):

$\frac{\partial f}{\partial y} = 14y\exp(y^2-x) = 14y\exp(y^2)\exp(-x)$

This is not bounded - right?

 

[LCKurtz]

That looks right.