# Need help for 2 interesting questions - rolle's and continuity

• Nov 23rd 2009, 08:06 PM
owq
Need help for 2 interesting questions - rolle's and continuity
1. Let f and g be functions such that f'' and g'' exist everywhere on R. For a < b, suppose that f(a)=f(b)=g(a)=g(b)=0, and $g''(x) \not= 0$ for every x $\in$ (a,b).

(i) Prove that $g(x) \not= 0$ for every x $\in$ (a,b). *I know how to do this, if I'm not wrong, use Rolle's Theorem a few times.*

(ii) Show that there exists a number c $\in$ (a,b) for which $\frac{f(c)}{g(c)} = \frac{f''(c)}{g''(c)}$ *No idea. Does it have something to do with the Mean Value Theorem?*

2. Let f be a continuous function on R such that $\lim_{x\to0}\frac{f(x)}{x}$ exists. Define the function
g(x) = $\int_0^1 f(xt) dt, x \in R$
Determine whether g' is continuous at x = 0. Justify your answer.
• Nov 24th 2009, 12:20 PM
Opalg
Quote:

Originally Posted by owq
1. Let f and g be functions such that f'' and g'' exist everywhere on R. For a < b, suppose that f(a)=f(b)=g(a)=g(b)=0, and $g''(x) \not= 0$ for every x $\in$ (a,b).

(i) Prove that $g(x) \not= 0$ for every x $\in$ (a,b). *I know how to do this, if I'm not wrong, use Rolle's Theorem a few times.*

(ii) Show that there exists a number c $\in$ (a,b) for which $\frac{f(c)}{g(c)} = \frac{f''(c)}{g''(c)}$

The function $f(x)g'(x) - f'(x)g(x)$ vanishes at both ends of the interval, so ..... (Rolle).