Answer to Question #314398 in Real Analysis for Pankaj

Question

Let f be a differentiable function on [α, β ] and x ∈[α, β ] .Show that, if

f ′(x) = 0 and , f ′′(x) >0 then f must have a local maximum at x.

Answer 1

$f''\left( x \right) >0\Rightarrow f''\left( t \right) >0,t\in \left( x-\varepsilon ,x+\varepsilon \right) ,\varepsilon >0\\Taylor‘s\,\,formula\,\,for\,\,\varDelta \in \left( -\varepsilon ,\varepsilon \right) :\\f\left( x+\varDelta \right) =f\left( x \right) +f'\left( x \right) \varDelta +\frac{1}{2}f''\left( \xi \right) \varDelta ^2,\xi \in \left( x-\varDelta ,x+\varDelta \right) \subset \left( x-\varepsilon ,x+\varepsilon \right) \Rightarrow \\\Rightarrow f\left( x+\varDelta \right) =f\left( x \right) +\frac{1}{2}f''\left( \xi \right) \varDelta ^2\geqslant f\left( x \right)$

Thus x is local minimum.