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

1 Answer
Jul 2, 2017

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.