Question

Is `f(x)=4x^5-12x^4+5x^2+2x+2` concave or convex at `x=-5`?

Answer 1

Concave

Explanation:

We use the second derivative to determine the curvature of a function. It is concave if the second derivative is less than zero and convex if the second derivative is greater than zero.

Weisstein, Eric W. "Second Derivative Test." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SecondDerivativeTest.html

`f(x)=4x^5−12x^4+5x^2+2x+2`

`f'(x)=20x^4−48x^3+10x+2`

`f''(x)=80x^3−144x^2+10`

`f''(-5) = 80(-5)^3−144(-5)^2+10`

`f''(-5) = -10000 − 3600 + 10 = -13590`

Definitely negative, so it is concave.