Mathematics 9709/11 — May/June 2013

Walkthrough

To decide whether a function is increasing, we look at its derivative. If $f'(x) > 0$ for every value in the domain, then the gradient of the curve is always positive, so the function is increasing.

First, differentiate $(2x-5)^3$. This is a composite function: the outer function is $u^3$ and the inner function is $u = 2x-5$. By the chain rule, the derivative is $3(2x-5)^2$ multiplied by the derivative of $2x-5$, which is $2$. So the derivative of $(2x-5)^3$ is $6(2x-5)^2$. Then add the derivative of $x$, which is $1$, giving

$$f'(x) = 6(2x-5)^2 + 1.$$

Now observe that $(2x-5)^2$ is a square, so it is always greater than or equal to zero. Multiplying by $6$ keeps it non-negative. Adding $1$ makes the whole expression strictly positive:

$$6(2x-5)^2 + 1 \ge 1 > 0.$$

Therefore the gradient is positive for every real $x$, so $f$ is increasing. It is important to argue generally, not just to test a few values.

Key Takeaways

• A function is increasing on an interval when its derivative is positive throughout that interval.
• The chain rule is essential for differentiating expressions like $(ax+b)^n$: multiply by the power, reduce the power by one, then multiply by the derivative of the inside.
• A squared term is always non-negative, which is often the key to proving that a derivative is positive.

Common Mistakes

• Forgetting to multiply by the derivative of the inner function $2x-5$, giving the incorrect derivative $3(2x-5)^2 + 1$.
• Substituting particular values of $x$ to "show" the function is increasing. The mark scheme explicitly says this is not accepted; you must show the derivative is positive for all $x$.
• Claiming the derivative is non-negative without noting that the $+1$ makes it strictly positive. While $\ge 0$ is allowed, the conclusion that $f$ is increasing follows most clearly from $f'(x) > 0$.

Things to Be Careful About

• The derivative should be written as $6(2x-5)^2 + 1$ or equivalently $24\left(x - \frac{5}{2}\right)^2 + 1$. Both forms make the positivity clear.
• The mark scheme awards one mark for $3(2x-5)^2$ and one mark for the multiplication by $2$ plus the $+1$. Make sure both parts are shown.
• The domain is all real numbers, so the argument must hold for every real $x$, not just for a restricted interval.