How do youfFind the value of c guaranteed by the mean value theorem for integrals #f(x) = - 4 / x^2# on the interval [ 1, 4 ]?

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Answer 1 · 2015-04-14T16:36:24

Solve the equation:

#f'(c) = (f(4) - f(1))/(4-1)#

The conclusion to the Mean Value Theoremtells us that there is such a #c# in the interval #(1,4)#

#f(4) = -1/4#

#f(1) = -4# and

#f'(x) = 8/x^3#

So you need to solve:

#8/c^3 = ((-1/4)-(-4))/(4-1) = (15/4)/3 = 15/12 = 5/4#

So we want:

#32/5 = c^3#

#c = root(3)(32/5)#

Now simplify and rationalize as your situation requires.