Question

For what values of x is `f(x)=(7x-1)(x-6)(x-2)` concave or convex?

Answer 1

`f(x)=(7x-1)(x-6)(x-2)` is concave down on the interval `x<19/7`, and `f(x)` is concave up on the interval `x>19/7`.

Explanation:

`f(x)=(7x-1)(x-6)(x-2)`
Find first derivative:
`f'(x)=21x^2-114x+92`
Find second derivative:
`f''(x)=42x-114`
`f''(x)=6(7x-19)`
Find where second derivative is negative to get intervals of concave down, and where `f''(x)` is positive is where `f(x)` is concave up.
`f''(x)` is positive when `x>19/7`, and `f''(x)` is negative when `x<19/7`.