Calculus Concavity and Derivative Tests

Question

What graph of y = f(x) would satisfy f’(-2) > 0 and f’’(2) > 0?

Answer 1

For f'(-2) to be greater than zero, f'(x) would have to be an equation with an even power. So, according to the process of the power rule, f(x) would have to have an odd power. This would also satisfy the condition of f''(2) greater than zero, since positive powers of positive numbers are always positive.

For example: y = x⁵

f'(x) = 5x⁴

f''(x) = 20x³

f'(-2) = 5(-2)⁴ = 80

f''(2) = 20(2)³ = 160