Find the average or mean slope of the function on this interval

Question

Consider the function f(x)=3x³-5x on the interval [-5,5]. Find the average or mean slope of the function on this interval.

By the mean value theorem, we know there exists at least on c in the open interval (-5,5) such that f^'(c) is equal to this mean slope. There are two values of c that work

what is the smaller one?

what is the larger one?

Answer 1

The mean slope of the function in this interval is given by

f(5)-f(-5)/(5-(-5))

f(5)= 3*125 - 25 = 350

f(-5)= - 3*125 + 25 = -350

then the average slope will be

(350-(-350))/10

=70

The points where it happens are:

9x²- 5 = 70 = f'(c)

X²= 25/3

X1 = (5*√3)/3

X2 = -(5*√3)/3

Answer 2

Here the derivative is 9x^2-5 and plugging in c makes that 9c^2-5=f(5)-f(-5) all divided by ten. 

F(5)=3x^3-5x or 375-25=350

F(-5)=-375+25

The difference is 700 making c the positive and negative square root of (700/10+5)/9,  the negative root being smaller.