# For what values of x is #f(x)= x-x^2e^-x # concave or convex?

##### 1 Answer

Find the second derivative and check its sign. It's convex if it's positive and concave if it's negative.

Concave for:

Convex for:

#### Explanation:

**First derivative:**

Take

**Second derivative:**

Now we must study the sign. We can switch the sign for easily solving the quadratic:

To make the quadratic a product:

Therefore:

- A value of
#x# between these two solutions gives a negative quadratic sign, while any other value of#x# makes it positive. - Any value of
#x# makes#e^-x# positive. - The negative sign at the start of the function reverses all signs.

Therefore,

Positive, therefore concave for:

Negative, therefore convex for: