Answer to Question #322672 in Real Analysis for Karan

Question

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

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

Answer 1

$f'\left( x \right) =0,f''\left( x \right) <0\\\\Since\,\,f'' is\,\,continuous\,\,on\,\,\left( \alpha ,\beta \right) , f'' is\,\,negative\,\,on\,\,some\,\,interval\,\,\left( x-\varepsilon ,x+\varepsilon \right) \\Taylor's\,\,formula\,\,for\,\,\left| \varDelta \right|<\varepsilon :\\f\left( x+\varDelta \right) =f\left( x \right) +f'\left( x \right) \varDelta +\frac{1}{2}f''\left( \xi \right) \varDelta ^2=f\left( x \right) +\frac{1}{2}f''\left( \xi \right) \varDelta ^2,\xi \,\,between\,\,x\,\,and\,\,x+\varDelta \\f''\left( t \right) <0,t\in \left( x-\varepsilon ,x+\varepsilon \right) \Rightarrow f''\left( \xi \right) <0\Rightarrow f\left( x+\varDelta \right) <f\left( x \right) \Rightarrow \\\Rightarrow x\,\,is\,\,a\,\,local\,\,\max \\$