Question

Let

f

be a differentiable function on

[Alpha, beta ]

and

x belongs to [alpha, beta ].

Show that, if

f '(x) = 0

and

f ''(x) =0,

then

f

must have a local maximum at

x.

Answer 1

As $x\in [\alpha,\beta]$

also f is a differentiable function

According to mean value theorem-

$f'(x)=\dfrac{f(\beta)-f(\alpha)}{\beta-\alpha}$

$f(\beta)-f(\alpha)=0\Rightarrow f(\beta)=f(\alpha)$

As f'(x) is 0 then, ,

There exist a critical point between $[\alpha,\beta]$

The function does not calls us nothing about the maximum or minimum.

As f''(x) =0, The given test does not tells nothings.

f(X) has local maximum at x.