Question

state and prove the necessary conditions for the existence of maximum value and minimum value?

Answer 1

The conditions for a function to be maximum or minimum include if

$f’ (x) = 0$;

and for maximum $f’’ (x)$ = negative

and for minimum $f’’ (x)$ = positive.

Assume that $f: (a,b) \to \R$

Theorem:

Let $c ∈ (a, b)$ and f be continuous at c. If for some $δ > 0,$ f has a minimum increasing on $(c − δ, c)$ and decreasing on $(c,c+δ),$ then f has a local maximum at c.

Proof:

Choose any $x_1$ and $x$ such that c $− δ < x1 < x < c.$ Then $f(x_1) ≤ f(x)$ and by the continuity of $f$ at $c$ we have $f(x_1) ≤ \lim x→c− f(x) = f(c).$

Similarly, if $c < x_2 < c + δ$ then $f(x_2) ≤ \lim x→c+ f(x) = f(c).$ This proves the result.