Review question

# What are the greatest and least values of $f(x) - g(x)$ on this interval? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R9056

## Solution

Functions $f$ and $g$ are defined by \begin{align*} f &\colon x \to x(x-1), \\ g &\colon x \to (x-1)(3x - 5), \end{align*}

where $x \in \mathbb{R}$ in each case.

1. Find the solution set, $S$, of the inequality $f(x) \ge g(x)$.
We have \begin{align*} f(x) &= x(x-1), \\ g(x) &= (x-1)(3x - 5). \end{align*}

Finding when $f(x) \ge g(x)$ is the same as finding when $f(x)-g(x)\ge0$.

We could expand the brackets, simplify and factorise, but instead notice that the two functions have a common factor…

\begin{align*} f(x)-g(x) &= x(x-1) - (x-1)(3x - 5) \\ &= (x-1)(-2x+5) \\ &= -2(x-1)\left(x-\tfrac{5}{2}\right) \end{align*}

So the graph of $y=f(x)-g(x)$ is a parabola with a maximum and $x$-intercepts at $x=1$ and $x=\frac{5}{2}$. $y$ is greater than zero between the two roots and the roots themselves are included in the interval.

$S$ is therefore the set of values $1\le x\le\frac{5}{2}.$

1. Sketch the graph of $y = f(x) - g(x)$ for $x \in S$, and state the greatest and least values of $f(x) - g(x)$ for $x \in S$.

We noted above that the graph of $y=f(x)-g(x)$ is a parabola with a maximum and zeros at $x=1$ and $x=\frac{5}{2}$. The minimum value of $y$ will be zero and the maximum value will occur half way between the two zeros, that is at $x=\frac{7}{4}$. $y_{max} = -2\left( \frac{7}{4}-1 \right)\left( \frac{7}{4}-\frac{5}{2} \right) = \frac{9}{8}$