SOLUTION: Verify that the given function f satisfies the hypotheses of the MVT on the given interval [a, b]. Then find all numbers c between a and b for which f(b) − f(a) b −

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Question 1075782: Verify that the given function f satisfies the hypotheses of the MVT on the given interval [a, b]. Then find all numbers c between a and b for which
f(b) − f(a)
b − a
= f '(c).
(Enter your answers as a comma-separated list. If the theorem does not hold, enter DNE.)
f(x) = 2x3 − 3x2 on [0, 4]
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
df%2Fdx=6x%5E2-6x=6x%28x-1%29
f%280%29=0
f%284%29=+2%284%29%5E3-3%284%29%5E2+=+2%2864%29-3%2816%29=80
So then,
df%2Fdx=%2880-0%29%2F%284-0%29=80%2F4=20
6x%5E2-6x=20
3x%5E2-3x-10=0
3%28x%5E2-x%29-10=0
3%28x%5E2-x%2B1%2F4%29-10=3%2F4
3%28x-1%2F2%29%5E2=3%2F4%2B40%2F4
3%28x-1%2F2%29%5E2=43%2F4
%28x-1%2F2%29%5E2=43%2F12
x-1%2F2=0+%2B-+sqrt%2843%2F12%29
x=1%2F2+%2B-+sqrt%2843%2F12%29
So only use the positive value,
x=1%2F2%2Bsqrt%2843%2F12%29
If we look at sqrt%2848%2F12%29=sqrt%284%29=2, so sqrt%2843%2F12%29%3C2
and
0%3Cx%3C2+%2B1%2F2
0%3Cx%3C5%2F2
So then,
0%3C=x%3C=4
So we have proven the MVT.