# How do we show that this is true?

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By inspection, I see an application of the Mean Value Theorem. Also, I know that #a < (a+b)/2 <b# for all #a,b in RR# ,

and since the polynomial f(x) is *well-behaved*, I do know that it is continuous and differentiable for all #x# , I can apply the MVT straightaway.

However, I don't know if this would be a 'rigorous' proof that I could write down in an exam. Please help me!

Thanks in advance :)

By inspection, I see an application of the Mean Value Theorem. Also, I know that

and since the polynomial f(x) is *well-behaved*, I do know that it is continuous and differentiable for all

However, I don't know if this would be a 'rigorous' proof that I could write down in an exam. Please help me!

Thanks in advance :)

##### 2 Answers

I do not think that the MVT is the good approach.

In fact, as

so that using the MVT we have:

but we cannot determine that necessarily:

We can however demonstrate the equality directly:

so that:

and:

Please see below.

#### Explanation:

The mean value theorem tells us that there is a solution to

Calculate

Calculate

Calculate