Given the function #f(x)=x^2/2-2x-1#, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [-1,1] and find the c?

Question

11026 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-02-22T07:38:38

The value of #c=0#

The mean value theorem states that if a function #f(x)# is

coontinuous interval #[a,b]# and differentiable on the interval #(a,b)#.

Then, there exists a point #c in (a,b)# such that

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

Here,

#f(x)=x^2/2-2x-1# is polynomial function which is continuous on the interval #[-1,1]# and differentiable on the interval #(-1,1)#

#f'(x)=x-2#

#f'(c)=c-2#

#f(-1)=1/2+2-1=3/2#

#f(1)=1/2-2-1=-5/2#

Therefore,

#(f(b)-f(a))/(b-a)=(f(1)-f(-1))/(1+1)=(-5/2-3/2)/2=-2#

#f'(c)=c-2=-2#

#c=0# and #c in (-1,1)#