Graphs of Functions and Differentiation

Differentiation is a powerful tool for determining the qualitative and quantitative features of a function's graph. While basic algebra tells us where a graph crosses the axes, calculus tells us about its turning points and general direction.

Stationary Points

A stationary point occurs when the tangent to a curve is horizontal. At such a point, the gradient is zero, meaning the rate of change of $y$ with respect to $x$ is null. Mathematically, we find stationary points by solving the equation:

$\frac{dy}{dx} = 0$

Once we have identified the $x$ values that satisfy this equation, we must classify the points as local maxima or local minima. There are several techniques for this.

Worked Example: Classifying Points on a Cubic

Consider the cubic function $y = 2x^3 - 3x^2 - 12x + 6$. To find the stationary points, we first differentiate the expression:

$\frac{dy}{dx} = 6x^2 - 6x - 12$

Setting this equal to zero and factorising:

$6(x^2 - x - 2) = 0$

$6(x - 2)(x + 1) = 0$

This gives us stationary points at $x = 2$ and $x = -1$. Because this cubic has a positive coefficient for the $x^3$ term, we expect the maximum to be on the left and the minimum on the right. Thus, the maximum occurs at $x = -1$ and the minimum at $x = 2$.

The Second Derivative Test

A more formal method for classification uses the second derivative, $\frac{d^2y}{dx^2}$, which tells us how the gradient itself is changing.

1. If $\frac{dy}{dx} = 0$ and $\frac{d^2y}{dx^2} > 0$, the point is a local minimum. This is because the gradient is increasing, turning from negative to positive.

1. If $\frac{dy}{dx} = 0$ and $\frac{d^2y}{dx^2} < 0$, the point is a local maximum. The gradient is decreasing, turning from positive to negative.

In our cubic example $y = 2x^3 - 3x^2 - 12x + 6$, the second derivative is $\frac{d^2y}{dx^2} = 12x - 6$.

At $x = 2$: $\frac{d^2y}{dx^2} = 12(2) - 6 = 18$. Since $18 > 0$, it is a minimum.

At $x = -1$: $\frac{d^2y}{dx^2} = 12(-1) - 6 = -18$. Since $-18 < 0$, it is a maximum.

Logical Nuances

It is important to note that the conditions $\frac{d^2y}{dx^2} > 0$ or $\frac{d^2y}{dx^2} < 0$ are sufficient but not necessary. For example, at the minimum of $y = x^4$, $\frac{dy}{dx} = 0$ and $\frac{d^2y}{dx^2} = 0$. This shows that if the second derivative is zero, the test is inconclusive and the point could still be a maximum, minimum, or a point of inflexion.

Increasing and Decreasing Functions

Calculus allows us to define precisely when a function is always moving "upwards" or "downwards" on a graph.

Strictly Increasing Functions

Intuitively, a strictly increasing function is one where the $y$ value always gets greater as the $x$ value gets greater. A sufficient condition for this using differentiation is:

If $f'(x) > 0$ for all $x$ in an interval, then the function is strictly increasing on that interval.

Strictly Decreasing Functions

A strictly decreasing function is one where the $y$ value always gets less as the $x$ value gets greater. The sufficient condition is:

If $f'(x) < 0$ for all $x$ in an interval, then the function is strictly decreasing on that interval.

Note that we cannot say the reverse: if a function is strictly increasing, it does not mean $f'(x) > 0$ for all $x$. Some functions may have corners where the derivative is not defined, yet they are still strictly increasing. However, for the smooth polynomial functions typical of the TMUA, looking for $f'(x) > 0$ or $f'(x) < 0$ is the primary method.

Points of Inflexion

While detailed examination of inflexions is excluded, you should have a qualitative understanding that they represent a point where the curve changes concavity. In the graph of $y = x^3$, the point at the origin is a horizontal point of inflexion because $f'(0) = 0$, but the graph does not turn back: it increases before and after the point.