Find the values of $x$ for which the graph of $f$ is concave-down. Justify your answer.

Write down the value of $a$.

Sketch the graph of $y=f\left(x\right)$ showing clearly the position of the points A and B.

Sketch the curve $y=f\left(x\right)$, 0 ≤ $x$ ≤ 5 indicating clearly the coordinates of the maximum and minimum points and any intercepts with the coordinate axes.

Sketch the graph of $x^2-y^2=1$, stating the coordinates of any axis intercepts and the equation of each asymptote.

Write down an expression for $y$ in terms of $x$.

Hence, show that $\int {\text{e}}^x\text{cos}{}^{2}x\text{d}x=\frac{{\text{e}}^x}{5}\text{sin}2x+\frac{{\text{e}}^x}{10}\text{cos}2x+\frac{{\text{e}}^x}{2}+c$,  $c\in \mathbb{R}$.

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

Find the value of $x$ for which $V$ is a maximum.

The coordinates of B can be expressed in the form B$\left(2^a,b\times 2^{-3a}\right)$ where a, b$\in \mathbb{Q}$. Find the value of a and the value of b.

Find an expression for $V$ in terms of $x$.

Use integration by parts to show that $\int {\text{e}}^x\text{cos}2x\text{d}x=\frac{2{\text{e}}^x}{5}\text{sin}2x+\frac{{\text{e}}^x}{5}\text{cos}2x+c$,  $c\in \mathbb{R}$.

Show that $g\left(x\right)$ satisfies the equation $g\text{'}\text{'}\left(x\right)=2(g\text{'}(x)-g(x\left)\right)$.

Hence determine the maximum value of $\frac{dP}{dt}$ in terms of $k$ and $N$.

find the value of $f\left(1\right)$.

Show that the volume of the cone may be expressed by $V=\frac{\pi }{3}\left(2R{h}^2-h^3\right)$.

Find the area enclosed by the curve and the $x$-axis between B and D, as shaded on the diagram.

Find the maximum volume.

Justify your answer.

Find $\frac{\text{d}V}{\text{d}x}$.

find the value of $f\left(4\right)$.

Given that there is one inscribed cone having a maximum volume, show that the volume of this cone is $\frac{32\pi R^3}{81}$.

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 minimum. 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 $f\left(0\right)$.

Show that ${\left(f\left(t\right)\right)}^2+{\left(g\left(t\right)\right)}^2=f\left(2t\right)$.

$f\left(\text{i}u\right)$.

The hyperbola with equation $x^2-y^2=1$ can be rotated to coincide with the curve defined by $xy=k, k\in \mathrm{ℝ}$.

Find the possible values of $k$.

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

Verify that $u=f\left(t\right)$ satisfies the differential equation $\frac{d^{2}u}{d{t}^2}=u$.

Show that ${\left(f\left(t\right)\right)}^2-{\left(g\left(t\right)\right)}^2={\left(f\left(\text{i}u\right)\right)}^2-{\left(g\left(\text{i}u\right)\right)}^2$.

Show that the graph of $y=q(x)$ is concave up for $x>1$.

Sketch the graph of $y=q(x)$ showing clearly any intercepts with the axes.

Find the $x$-coordinates of A and of C , giving your answers in the form $a+\text{arctan}b$, where $a$, $b\in \mathbb{R}$.

$g\left(\text{i}u\right)$.

Hence find, and simplify, an expression for ${\left(f\left(\text{i}u\right)\right)}^2+{\left(g\left(\text{i}u\right)\right)}^2$.