# #x^3+24x-16# [0,4] verify mean value theorem?

##### 1 Answer

May 2, 2018

We seek to verify the Mean Value Theorem for the function

# f(x) = x^3+24x-16# on the interval#[0,4]#

The Mean Value Theorem, tells us that if

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

So, Differentiating wrt

# f'(x) = 3x^2 + 24 #

And we seek a value

# :. 3c^2 + 24 = ((64+96-16)-(0+0-16))/(4-0) #

# :. 3c^2 + 24 = 160/4 #

# :. 3c^2 + 24 = 40 #

# :. 3c^2 = 16 #

# :. c^2 = 16/3 #

# :. c = +- (4sqrt(3))/3 #

And we require that