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

1 Answer
Steve M
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(x)# is differentiable on a interval #[a,b]# then #EE \ c in [a,b]# st:

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

So, Differentiating wrt #x# we have:

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

And we seek a value #c in [0,4]# st: # f'(c) = (f(4)-f(0))/(4-0) #

# :. 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 #c in [0,4]#, so we choose #c=(4sqrt(3))/3#

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