Problem 1 - Differentiable Functions

Let $f(x)$ be a thrice differentiable function with $f'(x)>0$. For a function $F(x)=f(x)/\sqrt{f'(x)}$, answer the following questions:

1.

Represent $F'(x)$ in terms of $f(x)$ and its derivatives.

2.

Represent $F''(x)$ in terms of $f(x)$ and its derivatives.

3.

Show that any crossing point of the graph $y=F(x)$ with the $x$-axis is a point of inflection.

1.

$$F'=\frac{f'\sqrt{f'} + f\frac{f''}{2f'\sqrt{f'}}}{f'}$$

$$=\sqrt{f'} + \frac{f f''}{2 {f'}^2 \sqrt{f'}}$$

2.

$$F''=-\frac{f''}{2f'\sqrt{f'}} + \frac{1}{2}\left( \frac{(f'f''+f ... ...\left[ 2f''f'\sqrt{f'} - \frac{{f'}^2f''}{2f'\sqrt{f'}}\right]}{{f'}^5} \right)$$

$$=-\frac{f''}{2f'\sqrt{f'}} + \frac{1}{2}\left( (f'f'' + f f'')\fr... ...\left[ \frac{2f''\sqrt{f'}}{{f'}^4}-\frac{f''}{2{f'}^4\sqrt{f'}}\right] \right)$$

3.

Let us assume that for some $x$, $F(x)=0$, i.e. $f(x)=0$. Then we have:

$$F'' = -\frac{f''}{2f'\sqrt{f'}} + \frac{1}{2}\left( f'f''\frac{\sqrt{f'}}{{f'}^3}\right)$$

$$=\frac{-f'f'' + f'' f'}{2{f'}^2\sqrt{f'}}$$

$$=0$$

That is, this point is a point of inflection.