Answer to Question #181387 in Real Analysis for kashish

Question

Suppose that y = f(x) : (−∞, ∞) → (−∞, ∞) is infinitely differentiable and has a local minimum at 0. Prove that there exists a disc centered on the y axis which lies above the graph of f and touches the graph at the point (0, f(0)).

Answer 1

Given function is-

$y=f(x):(-\infty,\infty)\to (-\infty,\infty)$

The given function is differentiable and have local minimum at x=0

So

$f'(x)=0, f(0)=0$

Also x=0 is the point of local minimum ao The value of $f''(x)>0,$ Hence The region must lies in the positive direction.

Also It exist at $x=0,$ The centered of the disc must lie on y-axis.

Hence There exist a disc centered on the y-axis which lies above the graph of f and touches the graph at the point (0, f(0)).