Verify that the hypotheses of the mean-value theorem are satisfied for f⁢(x)=x+1 on the interval [0,63], and find all values of c in that interval that satisfy the conclusion of the theorem. enter the exact answers in increasing order. if there is only one value of c, enter na in the second answer area. note that "c=" is already provided. do not include this in your submitted responses to this question. c=

The mean value theorem states that, given a function continuous in some interval and differentiable in the same open interval , there exists a point such that

In your case, the function is a polynomial, and as such it is continuous and differentiable infinite times on the whole real number set. So, in particular, it is continuous in and differentiable in the same open interval .

Moreover, the derivative of the function is

So, the statement of the mean value theorem becomes the following: there exists a point such that

But we know that is constantly 1, so we have

which is an identity, i.e. an equation which is always verified.

So, every point in satisfies the mean value theorem.

Why? Well, the theorem states the existance of a point such that the tangent to that point (which is [/tex] f'(c) [/tex]) has the same slope as the secant connecting and .

But in this case, is a line, which means that the said secant and the function itself are actually the same thing.

in the math equation 8 x 7 = 56, the 8 and the 7 are both factors of 56, since in this lesson we will use square tiles to figure out the factors of a whole find as many different arrays as you can, but follow one important rule: all the rows must have the i can make two rows of nine, so 2 and 9 are both factors of 18.

step-by-step explanation:

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step-by-step explanation: