Question

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Let g(x) = 1/(x + 1) (a) Does g satisfy the conditions of the Mean Value...

Let g(x) = 1/(x + 1)

(a) Does g satisfy the conditions of the Mean Value Theorem over the interval [1, 3]? Explain.

(b) Find a number c that satisfies the conclusion of the Mean Value Theorem for g over [1, 3], or explain why no such number exists.

(c) Does g satisfy the conditions of the Mean Value Theorem over the interval [−3, 0]? Explain.

(d) Find a number c that satisfies the conclusion of the Mean Value Theorem for g over [−3, 0], or explain why no such number exists.

Expert Solution

g(x) is undefined on the point x=-1 and defined everywhere else.

a) g(x) can be written as p(x)/q(x), where p(x) =1 and q(x) =1+x.

And in the region [1,3], p(x) and q(x) are both defined and are continuous and differentiable as p(x) is a constant and q(x) is a linear polynomial and q(x) is non-zero . So, g(x) is contiuous on[1,3] and also differentiable on(1,3).

So, g(x) satisfies both the conditions of mean value theorem on[1,3].

b) The boundary points are 3 and 1.

Therefore in order to find c, we can do:

For our range,

c) The range[-3,0] contains the point x=-1.

And the function g(x) is not defined at that point.

Therefore g(x) is not continuous in the given interval. So, it does not satisfy the condition of MVT.

d) Let's assume that MVT exists.

Therefore, to find c:

Therefore no real value of c exists.

Please comment in case of any confusion.

Happy to help.

milcah answered 2 years ago

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Let g(x) = 1/(x + 1) (a) Does g satisfy the conditions of the Mean Value...

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