1. ## True or False

If true, show or explain why. If false provide a counterexample ( that is give a specific instance where the statement is not true).

a.) Let functions f and g along with their second derivatives exist for all x. If f and g are both concave up for all x, then f+g is concave up for all x.

b.) (f g)' is never equal to f' * g'

c.) The only functions with the fourth derivative y''''= sin x must be of the form y = sin x + c where c is a constant.

d.) The nth derivative of the function y = a^x is y to nth derivative = a^x(lna)^n

2. Hello, erimat89!

Here's some help . . .

b) $(f g)'$ is never equal to $f'\cdot g'$
False

Let $f(x) \:=\:3x,\;\;g(x) = \frac{C}{1-x}$

Then: . $fg \;=\;\frac{3Cx}{1-x}$

And: . $(fg)' \;=\;3C\left[\frac{(1-x)\!\cdot\!1 - x(\text{-}1)}{(1-x)^2}\right] \quad\Rightarrow\quad (fg)' \;=\;\frac{3C}{(1-x)^2}$

We have: . $\begin{array}{ccccccc}f(x) &=& 3x & \Rightarrow & f'(x) &=& 3 \\ g(x) &=&\frac{C}{1-x} & \Rightarrow & g'(x) &=& \frac{C}{(1-x)^2} \end{array}\quad\Rightarrow\quad f'g' \:=\:\frac{3C}{(1-x)^2}$

c) The only functions with the fourth derivative $y''''\:=\: \sin x$ must be of the form: $y \:= \:\sin x + c$
False

The function could be: . $y \:=\:\sin x + 2x^3- 9$

d) The $n^{th}$ derivative of the function $y \:= \:a^x$ is: . $\frac{d^ny}{dx^n} \:=\:a^x(\ln a)^n$
True

$\begin{array}{ccc} y &=& a^x \\ \\[-3mm]
\dfrac{dy}{dx} &=& a^x(\ln a) \\ \\[-3mm]
\dfrac{d^2y}{dx^2} &=& a^x(\ln a)^2 \\ \\[-3mm]
\dfrac{d^3y}{dx^3} &=& a^x(\ln a)^3 \\
\vdots & & \vdots \end{array}$

$\begin{array}{ccc}\dfrac{d^ny}{dx^n} &=& a^x(\ln a)^n \end{array}$

3. ## Why is part b) false?

The example you given f'g' looks to me that it's equal to (f g)' with 3C/(1-x)^2 being the derivative of both functions whether its f'g' or (f g)'

4. ## I think i got it

Nevermind i think i understand thanks for your help