Stationary points and increasing and decreasing functionsEdexcel International A Level Maths: Revision notes

The derivative $\frac{dy}{dx}$ gives the gradient of the tangent. A stationary point is a point where $\frac{dy}{dx}=0$, so the tangent is horizontal. To find them: differentiate, set $\frac{dy}{dx}=0$, solve for $x$, then substitute each $x$ into the original equation to get the $y$-coordinate. Example: $y=x^3-6x^2+9x+2$ has $\frac{dy}{dx}=3x^2-12x+9=3(x-1)(x-3)$, so the stationary points are $(1,6)$ and $(3,2)$. A stationary point may be a local maximum, a local minimum or a point of inflection. Maxima and minima are also called turning points.

A function $\mathrm{f}$ is increasing on an interval if $\mathrm{f}'(x)>0$ for all $x$ in it, and decreasing if $\mathrm{f}'(x)<0$. To find where $\mathrm{f}$ is increasing or decreasing, solve the inequality $\mathrm{f}'(x)>0$ or $\mathrm{f}'(x)<0$. For a quadratic $\mathrm{f}'$, factorise and sketch it: a positive $x^2$ coefficient means it is negative between the roots and positive outside them. Example: $\mathrm{f}(x)=2x^3-3x^2-12x+5$ has $\mathrm{f}'(x)=6(x+1)(x-2)$. So $\mathrm{f}$ is decreasing for $-1<x<2$ and increasing for $x<-1$ or $x>2$.

Second derivative test. At a stationary point: if $\frac{d^2y}{dx^2}>0$ it is a local minimum; if $\frac{d^2y}{dx^2}<0$ it is a local maximum. If $\frac{d^2y}{dx^2}=0$ the test is inconclusive. Instead use the gradient test: find the sign of $\frac{dy}{dx}$ just before and just after the point. Maximum: $+$, $0$, $-$. Minimum: $-$, $0$, $+$. Point of inflection: the sign does not change. Example: $y=x^4$ has $\frac{dy}{dx}=4x^3$ and $\frac{d^2y}{dx^2}=12x^2=0$ at $x=0$. The gradient is negative for $x<0$ and positive for $x>0$, so $(0,0)$ is a minimum.

To sketch a curve from its equation: find where it crosses the axes (set $x=0$ for the $y$-intercept, $y=0$ for the roots); find the stationary points and their nature; and use the leading term for the behaviour as $x\to\pm\infty$. A cubic with a positive $x^3$ coefficient rises from bottom left to top right; with a local maximum and minimum it has an S-shape. The stationary points also count the roots of $\mathrm{f}(x)=k$: the number of times the horizontal line $y=k$ meets the curve. For a cubic with local maximum $12$ and local minimum $-15$, $\mathrm{f}(x)=k$ has three distinct roots only when $-15<k<12$.

$y=x^3-3x^2-9x+7$.

1. $\frac{dy}{dx}=3x^2-6x-9=3(x+1)(x-3)$, so $x=-1$ or $x=3$.
2. $y(-1)=-1-3+9+7=12$ and $y(3)=27-27-27+7=-20$.
3. $\frac{d^2y}{dx^2}=6x-6$: at $x=-1$ it is $-12<0$ (maximum); at $x=3$ it is $12>0$ (minimum).
4. Sketch: crosses the $y$-axis at $7$, rises to a maximum at $(-1,12)$, falls to a minimum at $(3,-20)$, then rises.
5. The curve is decreasing for $-1<x<3$.