Calculus (Year 12 Maths Advanced) quiz

$3x^2 - 4$

$\frac{x^4}{4} - 2x^2$

$3x^2$

$3x - 4$

$e^{2x}$

$2e^{2x}$

$x e^{2x}$

$\frac{e^{2x}}{2}$

$-\frac{1}{x^2} + C$ (for $x \neq 0$)

$\frac{x^0}{0} + C$

$\ln(x) + C$ (for $x > 0$)

$\frac{1}{x^2} + C$

A local minimum

A point of inflection

A discontinuity

A local maximum

$\ln(x^2 + 1) + C$

$2 \ln(x) + C$

$x^2 + 1 + C$

$\frac{2x}{x^2 + 1} + C$

$\frac{8}{3}$

$\frac{32}{3}$

$\frac{16}{3}$

$\frac{64}{3}$

$1$

$-1$

$0$

$\frac{\pi}{2}$

$2\pi$

$4\pi$

$\frac{16\pi}{3}$

$\frac{8\pi}{3}$

$0$

$9$

$27$

$18$

$x$

$\frac{1}{x}$

$e^x$

$\frac{1}{\ln(x)}$

$-3$

$0$

$1$

$4$

I will find the maximum value somehow.

Optimisation is basically just when you try to maximise or minimise something, and that is generally what this whole question is really about, so I am now going to have a go at doing exactly that in this answer.

First, I will draw a quick diagram.

Let $x$ be the variable I want to optimise. I will write the quantity to be optimised, differentiate, set the derivative to zero, and verify with the second derivative test.