I have an exam coming up tomorrow and this is one of the types of problems that

Question

I have an exam coming up tomorrow and this is one of the types of problems that I was having difficulty with. It would be great if I could get a step by step breakdown of how everything is solved and the reasoning for it, because I must have been absent during this lecture. I will award the points to the first person who can help me understand how to solve these problems. Thank you.

Please see image below for the problem:

For each of the following, use the intermediate value theorem to show that there exists a number C in [a, b] Such that f = M Clearly explain why can or Cannot us this theorem. f(x) = x2 - 4x+6 On [0, 3], M = 4 f(x) = 4/x+z on [-3, 1], M = 1/z

Explanation / Answer

A.)

The intermediate value theorem states that if a function is continuous on [a,b] and f(a) is opposite sign to f(b), then somewhere between a and b, there must be a value c, such that f(c) = 0

The problem you have is an extension of the intermediate value theorem.

Since f(x)=x^2 - 4x + 6 is continuous on [0,3] (All polynomials are continuous)

Then so is g(x)=f(x)-M=x^2-4x+2

g(0)=2

g(3)=-1

Since g(0) is opposite sign to g(3), somewhere between 0 and 3, there must be a point where g(x) = 0

And since that is the case, if g(x) equals 0 between 0 and 3, then f(x)=g(x)+4 must be equal to 4 somewhere between 0 and 3

The next problem, the intermediate value theorem does not apply, since the function is discontinuous at x=-2 which is between -3 and 1

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