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 1075783: 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) = 7 sin(x) + 7 cos(x) on [0, 2π]
Answer by htmentor(1343) (Show Source): You can put this solution on YOUR website!
f(x) = 7 sin(x) + 7 cos(x) on [0, 2π]
The average rate of change over the interval is:
(f(2π) - f(0))/(2π - 0) = (7 - 7)/2π = 0
Now we need to find c such that f'(c) = 0 over the interval
f'(x) = 7cos(x) - 7sin(x) = 7(cos(x) - sin(x)) = 0 -> cos(x) = sin(x), or tan(x) = 1
Thus we need find all values of c satisfying tan(c) = 1 on [0, 2π]
There are two such values:
c = π/4 and c = 5π/4
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