How do we show that this is true?
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
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