Does the function f(x)=#sqrtx# satisfy the hypothesis of the Mean Value Theorem on the closed interval [0,4]? Then find all numbers c that satisfy the conclusion of the Mean Value Theorem

#sqrtx#

1 Answer
Apr 7, 2018

#c=1#

Explanation:

The Mean Value Theorem tells us that if we have some function #f(x)# that is continuous on the interval #[a,b]# and differentiable on #(a,b),# then there is some number #c# in #(a,b)# such that #f'(c)=(f(b)-f(a))/(b-a)#

In other words, at some point, the derivative will be equal to the function's average rate of change.

So, we're given #f(x)=sqrtx#. It is continuous on #[0,4]# and differentiable on #(0,4)#, so these conditions are satisfied.

Take the first derivative:

#f'(x)=1/(2sqrtx)#

Adjust for #c:#

#f'(c)=1/(2sqrtc)#'

Equate to the average rate of change over the interval:

#1/(2sqrtc)=(f(4)-f(0))/(4-0)#

#1/(2sqrtc)=1/2#

#1/sqrtc=1#

#sqrtc=1#

#c=1#

#0<1<4,# so, this is a valid solution.