How do you find the number c that satisfies the conclusion of the Mean Value Theorem for the function #f(x)=x^3 - 2x + 1# on the interval [0,2]?

Answer 1

Solve #f'(x) = (f(2)-f(0))/(2-0)# Discard any solution(s) outside #(0,2)#.

The conclusion of the Mean Value Theorem for function #f# on interval #[a,b]# is
there is a number #c# in #(a,b)# such that #f'(c) = (f(b)-f(a))/(b-a)#
(Alternatively such that #f(b)-f(a)=f'(c)(b-a)# which is equivalent by algebra.)
So the #c# or #c#'s that satisfy the conclusion for this function on this interval are exactly the solutions to
#f'(x) = (f(2)-f(0))/(2-0)#
that are in #(0,2)#.
Since the solutions of #3x^2-2 = 2# are #+-sqrt(4/3)#, the #c# we want is #sqrt(4/3)#, which is better written as #(2sqrt3)/3#.
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Answer 2

To find the number c that satisfies the conclusion of the Mean Value Theorem for the function (f(x)=x^3 - 2x + 1) on the interval ([0,2]), first calculate the average rate of change of the function over the interval. This is done by finding the slope of the secant line between the endpoints of the interval.

The average rate of change is (\frac{f(2) - f(0)}{2 - 0}).

Next, find the derivative of the function, (f'(x)), which is (3x^2 - 2).

Then, set (f'(c)) equal to the average rate of change and solve for c. This gives the equation (f'(c) = \frac{f(2) - f(0)}{2 - 0}).

Substitute the function and its derivative into the equation and solve for c:

(3c^2 - 2 = \frac{((2)^3 - 2(2) + 1) - ((0)^3 - 2(0) + 1)}{2 - 0}).

Simplify the equation and solve 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.

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.

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.

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.

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