Find all the values of $x$ where the graph of $f$ is increasing. Justify your answer.

Find the value of $x$ where the graph of $f$ has a local maximum.

Find the value of $x$ where the graph of $f$ has a local minimum. Justify your answer.

Find the values of $x$ where the graph of $f$ has points of inflexion. Justify your answer.

The total area of the region enclosed by the graph of $f\text{'}$, the derivative of $f$, and the $x$-axis is $20$.

Given that $f\left(p\right)+f\left(t\right)=4$, find the value of $f\left(0\right)$.

Special note: In this question if candidates use the word 'gradient' in their reasoning. e.g. gradient is positive, it must be clear whether this is the gradient of $f$ or the gradient of $f\text{'}$ to earn the R mark.

$f$ increases when $p<x<0$                A1

$f$ increases when $f\text{'}\left(x\right)>0$   OR   $f\text{'}$ is above the $x$-axis                R1

Note: Do not award A0R1.

[2 marks]

Special note: In this question if candidates use the word 'gradient' in their reasoning. e.g. gradient is positive, it must be clear whether this is the gradient of $f$ or the gradient of $f\text{'}$ to earn the R mark.

$x=0$               A1

[1 mark]

Special note: In this question if candidates use the word 'gradient' in their reasoning. e.g. gradient is positive, it must be clear whether this is the gradient of $f$ or the gradient of $f\text{'}$ to earn the R mark.

$f$ is minimum when $x=p$               A1

because $f\text{'}\left(p\right)=0, f\text{'}\left(x\right)<0$ when $x<p$ and $f\text{'}\left(x\right)>0$ when $x>p$

(may be seen in a sign diagram clearly labelled as $f\text{'}$)

OR because $f\text{'}$ changes from negative to positive at $x=p$

OR $f\text{'}\left(p\right)=0$ and slope of $f\text{'}$ is positive at $x=p$               R1

Note: Do not award A0 R1

[2 marks]

Special note: In this question if candidates use the word 'gradient' in their reasoning. e.g. gradient is positive, it must be clear whether this is the gradient of $f$ or the gradient of $f\text{'}$ to earn the R mark.

$f$ has points of inflexion when $x=q, x=r$ and $x=t$               A2

$f\text{'}$ has turning points at $x=q, x=r$ and $x=t$

OR

$f\text{'}\text{'}\left(q\right)=0, f\text{'}\text{'}\left(r\right)=0$ and $f\text{'}\text{'}\left(t\right)=0$ and $f\text{'}$ changes from increasing to decreasing or vice versa at each of these $x$-values (may be seen in a sign diagram clearly labelled as $f\text{'}\text{'}$ and $f\text{'}$)              R1

Note: Award A0 if any incorrect answers are given. Do not award A0R1

[3 marks]

Special note: In this question if candidates use the word 'gradient' in their reasoning. e.g. gradient is positive, it must be clear whether this is the gradient of $f$ or the gradient of $f\text{'}$ to earn the R mark.

recognizing area from $p$ to $t$ (seen anywhere)               M1

$\underset{p}{\overset{t}{\int }}\left|f\text{'}\left(x\right)\right|dx$

recognizing to negate integral for area below $x$-axis               (M1)

$\underset{p}{\overset{0}{\int }}f\text{'}\left(x\right)dx-\underset{0}{\overset{t}{\int }}f\text{'}\left(x\right)dx$   OR   $\underset{p}{\overset{0}{\int }}f\text{'}\left(x\right)dx+\underset{t}{\overset{0}{\int }}f\text{'}\left(x\right)dx$

$\underset{m}{\overset{n}{\int }}f\text{'}\left(x\right)dx=f\left(n\right)-f\left(m\right)$ (for any integral)               (M1)

$f\left(0\right)-f\left(p\right)-\left[f\left(t\right)-f\left(0\right)\right]$   OR   $f\left(0\right)-f\left(p\right)+f\left(0\right)-f\left(t\right)$               (A1)

$2f\left(0\right)-\left[f\left(t\right)+f\left(p\right)\right]=20, 2f\left(0\right)-4=20$               (A1)

$f\left(0\right)=12$               A1

[6 marks]