Is f(x)=(x-2)^3-x^4+x concave or convex at x=0?

1 Answer
Sep 14, 2017

take the second derivative...

Explanation:

f'(x) = 3(x-2)^2 - 4x^3 + 1

f''(x) = 6(x-2) - 12x^2

...evaluate f''(x) at x = 0:

= 6(-2) = -12

So this means at the slopes of the tangent lines to the original function are decreasing (as x increases) at this point.

So, I'd say that the function is "concave downward" at this point. That would make it "Convex upward".

A graph of the original function is helpful:

graph{(x-2)^3 - x^4 + x [-10, 10, -5, 5]}