Determine the interval(s) at which f(x) is concave up given that f′′(x)=−x2+x+6.

Question

Critical points are: -2 & 3

Answer 1

To be concave up the 2nd derivative must be positive, so

-x^2 + x + 6 > 0
x^2 - x - 6 < 0
(x-3)(x+2) < 0
looking at y = x^2 - x - 6 , we see from our solution above that it has x-intercepts of 3 and -2, thus it is above ( greater than zero) for
x < -2 or x > 3