Assume that the only solution to f" (x) = 0 is x = 2 and the only solutions to f' (x) x = 1 and x = 3. Which of the following need NOT be true: f is decreasing for 1 < x < 3 and is increasing otherwise f is concave down for r < 2 and concave up for r> 2 O f is increasing for r < 2 and decreasing for x > 2 f has a unique local maximum at r = 1 that is not an absolute maximum

Given that $f\to \infty$ as $x\to \infty$ and $f\to -\infty$ as $x\to -\infty$ and the solution for $f\text{'}\text{'}=0$ is $x=2$ and the only solution for $f\text{'}\left(x\right)$ is $x=1$ and $x=3$.

From the information we have $f\text{'}\left(x\right)=(x-1)(x-3)$ and also $f\text{'}\text{'}=2x-4$ satisfies all the properties from these 

$\begin{array}{rcl}f& =& \int \left(x-1\right)\left(x-3\right)\\ & =& \int x^2-4x+3\\ & =& \frac{x^3}{3}-2x^2+3x+c\end{array}$

Without loss of generality let's assume c=0