Question

Is `f(x) =4xe^x-3x^2` concave or convex at `x=-1`?

Answer 1

Concave (also called concave down).

Explanation:

To determine concavity and convexity, we look at the sign of the second derivative.

• If `f''(-1)<0`, then `f(x)` is concave at `x=-1`.
• If `f''(-1)>0`, then `f(x)` is convex at `x=-1`.

To find the first derivative, use the product rule on the `4xe^x` term.

`f(x)=4xe^x-3x^2`

`f'(x)=4e^x+4xe^x-6x`

`f''(x)=4e^x+4e^x+4xe^x-6=4xe^x+8e^x-6`

Find the sign of the second derivative at `x=-1`.

`f''(-1)=-4e^-1+8e^-1-6=4/e-6`

Since this is `>0`, the function is concave at `x=-1`. This means its shape is similar to that of `nn`. We can check a graph of the original function:

graph{4xe^x-3x^2 [-3, 3, -15, 15]}