The Mean Value Theorem applies to the given function on the given interval. How do you find all possible values of #f(x)=x^(9/4)# on the interval [0,1]?

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Answer 1 · 2017-10-08T05:49:57

The value of #c=0.44#

The Mean Value Theorem states that :

If #f(x)# is a continuous function on the interval #[a,b]# and differentiable on the interval #(a,b)#.

Then , #EE c in (a,b)# such that

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

Here,

#f(x)=x^(9/4)#

This function is continuous on the interval #[0,1]# and differentiable on the interval #(0,1)#

#f(0)=0#

#f(1)=1#

#f'(x)=9/4x^(9/4-1)=9/4x^(5/4)#

Therefore,

#9/4c^(5/4)=(f(1)-f(0))/(1-0)=(1-0)/(1)=1#

#c^(5/4)=4/9#

Taking the natural logs on both sides

#5/4lnc=ln(4/9)#

#lnc=4/5ln(4/9)=-0.81#

#c=0.44#

And #c in (0,1)#

graph{x^(9/4) [-1.698, 2.626, -0.227, 1.936]}