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.