How do you determine whether the function f(x)=2x^3-3x^2-12x+1 is concave up or concave down and its intervals?

1 Answer
Aug 19, 2015

Investigate the sign of the second derivative.
f is concave down on (-oo, 1/2) and concave up on (1/2, oo).
(1/2, -11/2) is the inflection point for the graph of f.

Explanation:

f(x)=2x^3-3x^2-12x+1

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

f''(x)= 12x - 6

f''(x) is never undefined and is 0 only at x=1/2

So the only x value at which the concavity might change is 1/2

For x < 1/2 we find that f''(x) < 0 and for x > 1/2, f''(x) > 0

So the graph of f is concave down on (-oo, 1/2) and concave up on (1/2, oo).

f(1/2) = 2(1/2)^3-3(1/2)^2-12(1/2)+1 = -11/2

(1/2, -11/2) is the only inflection point for the graph of f.