Consider the function #f(x) = 4 x^3 − 4 x# on the interval [ −3 , 3 ], how do you find the average or mean slope of the function on this interval?

1 Answer
Oct 26, 2017

See the explanation below

Explanation:

The average or mean slope of a function #f(x)# on the interval #[a,b]# is

#=(f(b)-f(a))/(b-a)#

Here,

#f(x)=4x^3-4x=4x(x^2-1)#

and the interval is #=[-3,3]#

#f(3)=12(8)=96#

#f(-3)=-12*8=-96#

Therefore,

The mean slope is

#=(f(3)-f(3))/(3-(-3))=(96+96)/(6)=32#

Then,

There exists #c in (-3,3)# such that

#f'(c)=32#

#f'(x)=12x^2-4=4(3x^2-1)#

Therefore,

#f'(c)=4(3c^2-1)=32#

#3c^2-1=8#

#c^2=9/3=3#

#c=+-sqrt3#

So,

#+-sqrt3 in (-3,3)#