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How do you find all numbers c that satisfy the conclusion of the Mean Value Theorem for #f(x)= x^3 + x - 1# over [0,2]?

Answer 1

First, find the derivative: #f'(x)=3x^2+1#. Next, find the average rate of change of #f# over the interval #[0,2]#: #\frac{f(2)-f(0)}{2-0}=\frac{10}{2}=5#. At this point, set #f'(c)=5# and solve for #c# as follows: #3c^{2}+1=5# so #3c^{2}=4# and #c^{2}=\frac{4}{3}#. There's one value of #c# between 0 and 2 that satisfies the conclusion of the Mean Value Theorem: #c=\sqrt{4/3}=\sqrt{4}/\sqrt{3}=2/\sqrt{3}=\frac{2\sqrt{3}}{3}#.

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Answer 2

Christopher Allers

To find all numbers ( c ) that satisfy the conclusion of the Mean Value Theorem for ( f(x) = x^3 + x - 1 ) over ([0, 2]), you first calculate the average rate of change of ( f ) over the interval. This is given by:

[ f'(c) = \frac{f(2) - f(0)}{2 - 0} ]

Calculate ( f(2) ) and ( f(0) ) and their difference. Then, find ( f'(c) ) by solving the equation for ( c ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.