Question

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

Answer 1

Convex.

Explanation:

Concavity is determined by the second derivative (positive = convex, negative = concave). So we need to find the second derivative of `f(x)`:
`f(x)=e^(3x)-2xlnx`
`f'(x)=3e^(3x)-2lnx-(2x)/x`
`f'(x)=3e^(3x)-2lnx-2`
`f''(x)=9e^(3x)-2/x`

To find the concavity at `x=1`, just plug `1` into `f''(x)`:
`f''(1)=9e^3-2`

This is positive, so the function is convex.