For what values of x is f(x)=(7x-1)(x-6)(x-2)f(x)=(7x1)(x6)(x2) concave or convex?

1 Answer
Nov 4, 2016

f(x)=(7x-1)(x-6)(x-2)f(x)=(7x1)(x6)(x2) is concave down on the interval x<19/7x<197, and f(x)f(x) is concave up on the interval x>19/7x>197.

Explanation:

f(x)=(7x-1)(x-6)(x-2)f(x)=(7x1)(x6)(x2)
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.