Question

Is `f(x)=3/x-2x` concave or convex at `x=9/4`?

Answer 1

Convex

Explanation:

Find the value of the second derivative at `x=9/4`. If the value is positive, then the function is convex. If it's negative, the function is concave at that point.

To find the function's second derivative, recall that `3/x=3x^-1` and then apply the power rule.

`f(x)=3x^-1-2x`

`f'(x)=-3x^-2-2`

`f''(x)=6x^-3=6/x^3`

The value of the second derivative at `x=9/4` is

`f''(9/4)=6/(9/4)^3`

This could be simplified, but it is evidently positive. Thus the function is convex at `x=9/4`.