Review question

Ref: R7433

## Solution

The figure shows part of a curve passing through $6$ points $A,B,C,D,E,F$.

Copy and fill in the following table showing, for each of the points, whether

1. $\dfrac{dy}{dx}$ is positive ($+$), negative ($-$), or zero ($0$),

2. $\dfrac{dy}{dx}$ is increasing ($I$), or decreasing ($D$), as $x$ increases.

The filled in table should look like this:

$A$ $B$ $C$ $D$ $E$ $F$
$\dfrac{dy}{dx}$ is $+,-,$ or $0$ $+$ $0$ $-$ $-$ $0$ $+$
$\dfrac{dy}{dx}$ is $I$ or $D$ $D$ $D$ $D$ $I$ $I$ $I$
1. To see whether $\dfrac{dy}{dx}$ is positive, negative or zero, we can imagine the tangent to the curve at each point and think about its gradient.

2. If we imagine the tangent to the curve at a point and move the point along the curve from left to right, we can see that its gradient starts positive and decreases. At $B$ it changes from positive to negative and is still decreasing. There is a point somewhere between $C$ and $D$ where the gradient stops decreasing and starts increasing.

Determine which of (a) and which of (b) apply at the point $\left(3,3\tfrac{1}{3}\right)$ on the curve $y=x+\dfrac{1}{x}$.

The derivative of $y=x+\dfrac{1}{x}$ is $\frac{dy}{dx}=1-\frac{1}{x^2} ,$ which is positive at $x=3$.

There are many ways to find if $\dfrac{dy}{dx}$ is increasing or decreasing, here are three ways you might determine it.

1. Calculation. We could evaluate $\dfrac{dy}{dx}$ at values of $x$ above and below $3$ and compare them:
$x=$ $2$ $3$ $4$
$\dfrac{dy}{dx}=$ $\dfrac{3}{4}$ $\dfrac{8}{9}$ $\dfrac{15}{16}$

We can see that the value of $\dfrac{dy}{dx}$ increases as $x$ increases.

2. Algebra. We could rewrite $\dfrac{dy}{dx}=\dfrac{x^2-1}{x^2}$. This makes it clear that for $x\geq 1$ $\dfrac{dy}{dx}>0$ and as $x$ gets larger $\dfrac{dy}{dx}$ gets closer and closer to $1$, so $y$ is increasing at $x=3$.

3. Graph sketching. A sketch of the graph $y=1-\dfrac{1}{x^2}$ shows that $y$ is increasing at $x=3$. And therefore the gradient of $y=x+\dfrac{1}{x}$ is increasing.

Can you find any other ways to determine if it is increasing or decreasing?