Mean Value Theorem
This theorem helps us to link monotonicity of a function with values of its derivative.
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Study Mean Value Theorem
Now we present the famous mean value theorem.
Theorem 1. Let \(f:[a,b]\to\mathbb{R}\) be differentiable. Then there exists some \(\hat{x}\in(a,b)\) such that \[f(b)-f(a)=f'(\hat{x})\cdot (b-a).\]
Before the proof is presented, we give some graphical interpretation: A division of the above equation by \(b-a\) gives \[\frac{f(b)-f(a)}{b-a}=f'(\hat{x}).\] The quantity on the left hand side is equal to the slope of the secant of \(f\) through \(a\) and \(b\), whereas \(f'(\hat{x})\) corresponds to the slope of tangent of \(f\) at \(\hat{x}\). Therefore, the secant of \(f\) through \(a\) and \(b\) is parallel to a tangent of \(f\).
Proof: Consider the function \(F:[a,b]\to\mathbb{R}\) with \[F(x):=f(x)-f(a)-\frac{f(b)-f(a)}{b-a}\cdot(x-a).\] Then we have \(F(a)=F(b)=0\). By Rolle’s Theorem, we get that there exists some \(\hat{x}\in(a,b)\) with \[0=F'(\hat{x})=f'(\hat{x})-\frac{f(b)-f(a)}{b-a}\] and thus \[f'(\hat{x})=\frac{f(b)-f(a)}{b-a}.\] \(\Box\)
The mean value theorem leads us to determine monotonicity properties of a function by means of its derivative.
Theorem 2. Let \(f:[a,b]\to\mathbb{R}\) be a differentiable function. Then the following holds true.
If \(f'(x)>0\) for all \(x\in(a,b)\), then \(f\) is strictly monotonically increasing.
If \(f'(x)<0\) for all \(x\in(a,b)\), then \(f\) is strictly monotonically decreasing.
If \(f'(x)\geq0\) for all \(x\in(a,b)\), then \(f\) is monotonically increasing.
If \(f'(x)\leq0\) for all \(x\in(a,b)\), then \(f\) is monotonically decreasing.
Proof: (i) By the mean value theorem we have that for \(x_1,x_2\in(a,b)\) with \(x_1<x_2\), there exists some \(\hat{x}\in(x_1,x_2)\) with \[f(x_2)-f(x_1)=f'(\hat{x})\cdot(x_2-x_1)>0.\] The results (ii)-(iv) can be proven analogously.\(\Box\)
An episode by Fabian Gabel and Julian Großmann.
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