# Thread: Derivatives and the Intermediate Value Property

1. ## Derivatives and the Intermediate Value Property

(1) Let $\displaystyle g_a(x)=\left\{\begin{array}{cc}x^{a},&\mbox{ if } x\geq 0\\0, & \mbox{ if } x<0\end{array}\right.$

(a) For which of the values of $\displaystyle a$ is $\displaystyle f$ continuous at zero?

(b) For which values of $\displaystyle a$ is $\displaystyle f$ differentiable at zero? In this case, is the derivative function continuous?

(c) For which values of $\displaystyle a$ is $\displaystyle f$ twice-differentiable?

(2) Let $\displaystyle f$ and $\displaystyle g$ be functions defined on an interval $\displaystyle A$, and assume both are differentiable at some point $\displaystyle c \in A.$

Prove the following:

(a) $\displaystyle (f+g)'(c) = f'(c) + g'(c)$

(b) $\displaystyle (kf)'(c) = kf'(c),$ for all $\displaystyle k \in R$ (R is the reals)

Any help would be greatly appreciated, thanks!

(1) Let $\displaystyle g_a(x)=\left\{\begin{array}{cc}x^{a},&\mbox{ if } x\geq 0\\0, & \mbox{ if } x<0\end{array}\right.$

(a) For which of the values of $\displaystyle a$ is $\displaystyle f$ continuous at zero?

(b) For which values of $\displaystyle a$ is $\displaystyle f$ differentiable at zero? In this case, is the derivative function continuous?

(c) For which values of $\displaystyle a$ is $\displaystyle f$ twice-differentiable?

(2) Let $\displaystyle f$ and $\displaystyle g$ be functions defined on an interval $\displaystyle A$, and assume both are differentiable at some point $\displaystyle c \in A.$

Prove the following:

(a) $\displaystyle (f+g)'(c) = f'(c) + g'(c)$

(b) $\displaystyle (kf)'(c) = kf'(c),$ for all $\displaystyle k \in R$ (R is the reals)

Any help would be greatly appreciated, thanks!

Here are a few hints to get you started.

use the limit def of continity

$\displaystyle f(0)=\lim_{x \to 0^+}f(x)$

use the differnce quotent for the derivative

$\displaystyle f'(0)=\lim_{x \to 0}\frac{f(x)-f(0)}{x-0}=\lim_{x \to 0}\frac{x^a-0}{x-0}$

Find the values of a so that the limit exits. Try to use a similar process to finish the problem.

3. Sorry, I still don't understand how to do problem 1. I figured out problem 2, that was easy. If anyone can help me out, thanks!

(1) Let $\displaystyle g_a(x)=\left\{\begin{array}{cc}x^{a},&\mbox{ if } x\geq 0\\0, & \mbox{ if } x<0\end{array}\right.$

(a) For which of the values of $\displaystyle a$ is $\displaystyle f$ continuous at zero?

(b) For which values of $\displaystyle a$ is $\displaystyle f$ differentiable at zero? In this case, is the derivative function continuous?

(c) For which values of $\displaystyle a$ is $\displaystyle f$ twice-differentiable?

(2) Let $\displaystyle f$ and $\displaystyle g$ be functions defined on an interval $\displaystyle A$, and assume both are differentiable at some point $\displaystyle c \in A.$

Prove the following:

(a) $\displaystyle (f+g)'(c) = f'(c) + g'(c)$

(b) $\displaystyle (kf)'(c) = kf'(c),$ for all $\displaystyle k \in R$ (R is the reals)

Any help would be greatly appreciated, thanks!
a) none, for that value of a is $\displaystyle x^a=0$
b) what value of a is $\displaystyle ax^{a-1}=0$
c)what about $\displaystyle a(a-1)x^{a-2}$

For two, use the difference quotient

5. Wow, I'm an idiot. lol Thanks!