12.20

Definition

We say \(c\) is a stationary point of \(f\) if \(f'(c)=0\)

Criterium for local maximum and minimum for \(f\)

Let \(f\) be differentiable function in \((a,b)\) and \(c\) be a stationary point such that \(c\in(a,b)\), then

• If \(f'(x)>0,\forall x\in (a,c)\) and \(f^{\prime}(x)<0,\forall x\in(c,b)\Rightarrow\) \(f\) has a local maximum in \(x=c\)​

Proof

Since \(f'(x)>0,\forall x\in(a,c)\), then \(f\) is strictly increasing \(\forall x\in(a,c)\Rightarrow f(x)\leq f(c),\forall x\in\left(a,c\right)\)​

Since \(f^{\prime}(x)<0,\forall x\in(c,b)\), then \(f\) is strictly decreasing \(\forall x\in(c,b)\Rightarrow f(x)\geq f(c),\forall x\in\left(c,b\right)\)

Then \(f(x)\leq f(c),\forall x\in (a,b)\), then \(f\) has a local maximum in \(x=c\) * If \(f'(x)<0,\forall x\in (a,c)\) and \(f^{\prime}(x)>0,\forall x\in(c,b)\Rightarrow\) \(f\) has a local minimum in \(x=c\)

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