Asked by Natalie
Find a point c satisfying the conclusion of the Mean Value Theorem for the function f(x)= x^1/3 on the interval [1,8]
I got f'(c)= 1/7 but am not sure where to go from there.
I got f'(c)= 1/7 but am not sure where to go from there.
Answers
Answered by
bobpursley
f'(x)=1/3 (x)^-2/3
f (1)=1^1/3=1
f (8) = 2
so, the point c must be such that
f'(c)= (f(8)-f(1))/(8-1)=(2-1)/7=1/7
but f'(c)=1/3 (c)^-2/3 and f'(c)=1/7 so
1/7=1/3(c^-2/3)
3/7=c^-2/3
take each side to the 3/2 power
c^2/3=7/3
c= cube root (7/3)^2 = 1.76
so, the conclusion of the theorem is borne out.
f (1)=1^1/3=1
f (8) = 2
so, the point c must be such that
f'(c)= (f(8)-f(1))/(8-1)=(2-1)/7=1/7
but f'(c)=1/3 (c)^-2/3 and f'(c)=1/7 so
1/7=1/3(c^-2/3)
3/7=c^-2/3
take each side to the 3/2 power
c^2/3=7/3
c= cube root (7/3)^2 = 1.76
so, the conclusion of the theorem is borne out.
There are no AI answers yet. The ability to request AI answers is coming soon!
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.