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

Question

Consider the function f(x)=6-7x² on the interval [-4,5].  Find the average or mean slope of the function on this interval

(f(5)-f(-4))/(5-(-4))=

By the Mean Value Theorem, we know there exists a c in the open interval (-4,5) such that f^'(c) is equal to this mean slope. For this problem, there is only one c that works. Find it.

Answer 1

The Mean Value Thm states:  If f(x) is continuous over [a, b], and f'(x) is defined over (a, b), where a<<b, then there is a c such that f'(c) = (f(b) - f(a))/(b - a).  Try this and you should have your answer.

 

Therefore:  f'(c) = -14c,  and (f(5)-f(4))/(5-(-4)) = (-169-(-106))/(9) = -63/9 = -7.

 

So -14c = -7, so c = 2.

 

I hope this helps.