1. concave

For what values of x is the graph of y=xe^−4x concave down?

2. Originally Posted by cwarzecha
For what values of x is the graph of y=xe^−4x concave down?
The x values for which the second derivative is positive.

3. ok sorry i should've been more specific. i don't know how to find the second derivative of that.

4. Originally Posted by cwarzecha
For what values of x is the graph of y=xe^?4x concave down?
Use second derivative test to see where the function is concave or convex.

5. whats the second derivative test?

6. Originally Posted by cwarzecha
For what values of x is the graph of y=xe^−4x concave down?
We have a product of functions, so we use the product rule.

$\displaystyle \frac{dy}{dx} = e^{-4x} - 4xe^{-4x} = e^{-4x}(1 - 4x)$.

$\displaystyle \frac{d^2y}{dx^2} = -4e^{-4x} - 4e^{-4x}(1 - 4x) = -4e^{-4x}(2 - 4x)$.

We require that this be greater than 0.

Notice that $\displaystyle -4e^{-4x}$ is always negative.

So for the second derivative to be positive, $\displaystyle 2 - 4x$ must also be negative (as negative times negative is positive).

$\displaystyle 2 - 4x < 0$

$\displaystyle 2 < 4x$

$\displaystyle x > \frac{1}{2}$.

So the values of x for which the graph is concave down are all those that are greater than $\displaystyle \frac{1}{2}$.