Consider the function f(x)=4x^3−6x on the interval [−4,4]. Find the average or mean slope of the function on this interval.

Question

Consider the function f(x)=4x^3−6x on the interval [−4,4]. Find the average or mean slope of the function on this interval.______________ 

By the Mean Value Theorem, we know there exists at least one c in the open interval (−4,4) such that f'(c) is equal to this mean slope.

For this problem, there are two values of c that work.

The smaller one is:__________ 

and the larger one is:_________ 

Answer 1

f(-4) = 4(-4)³ - 6(-4) = 4(-64) - (-24) = -232

f(4) = 4(4)³ - 6(4) = 256 - 24 = 232

f'(c) = (232 - (-232)) / (4 - (-4)) = 464 / 8 = 58

f(c) = 4x³ - 6x

f'(c) = 12x² - 6

58 = 12x² - 6

64 = 12x²

64/12 = x²

16/3 = x²

x = ±√16/3

The MVT is satisfied at c = ±√16/3