Find the set of all x-values when f(x)= 2x4-3x2-20x+7 is concave down

Question

Find the set of all x-values when f(x)= 2x⁴-3x²-20x+7 is concave down

Answer 1

It will be concave down when f''(x) < 0

f'(x) = 8x³ - 6x - 20

f'' (x) = 24x² - 6

f'' (x) is a quadratic, facing up (a>0) with zeroes when 24x² = 6 ie x = +1/2, -1/2

f''(x) will be negative when -1/2 < x < + 1/2, that is when f(x) is concave down

Answer 2

The derivative is equal to f'(x)=8x^{3}-6x-20 and so the second derivative equals f''(x)=24x^{2}-6. For concave down we need to solve the inequality 24x^{2}<=6 which is the same as x^{2}<=1/4 or |x|<=1/2. So the desired interval is (-1/2,1/2).