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Maxima and minima

The derivative of a function provides useful information about the shape of the graph of the function. When the derivative is positive, the graph is sloping upwards. When the derivative is negative, the graph is sloping downwards.

The derivative f(x) is a function that gives the slope of the graph of f at any value of x, provided the curve is smooth (a tangent exists) at the value of x.

Newton's notation: $f'(x)$

Leibniz's notation. ${dy}/{dx}$

At the point where the derivative changes from positive to negative, there is a 'stationary point'. Provided the function is continuous, the stationary point could reverse the gradient of the tangent, and become a turning point.

Maximum and minimum of a function
Maximum and minimum of a function: finding the point where the slope changes direction

Turning points are where the derivative is equal to zero. At that point, the tangent is parallel to the x-axis. But how do we tell if this point is at the bottom (minimum) or top (maximum) of a curve?

To so this we create a 'sign chart'. A sign chart is a line with the turning point marked, and the gradient of the tangent on either side indicated.

The Sign Chart

The sign chart for $f(x) = x^3 - 3x$ would be:

x< -1-1> -1< 11> 1
$f'(x)$> 00< 0< 00> 0

From the table it is evident that as the curve approaches the first turning point (-1, 2), it is sloping upwards ($f'(x) > 0$), and is sloping downwards past the turning point ($f'(x) < 0$), so this turning point is a maximum. In this case of a parabola, we say it is concave up.

The turning points are not necessarily the absolute highest or lowest values for the curve, so they are called relative (or local) maxima or minima.

Maximum, minimum and inflexion point
Maximum, minimum and inflexion point

Inflection Points

By taking a second derivative ($f″(x)$), we can learn something else about the curve: the first derivative tells us the slope of the tangent, or the rate of change of $f(x)$. Similarly, the second derivative tells us the rate of change of the first derivative.

For example, the first derivative of a displacement equation gives us the rate of change of position, or velocity. The second derivative will give us the rate of change of velocity, or the acceleration.

When the second derivative is zero ($f''(x) = 0$), the first derivative changes its slope from positive to negative, or from negative to positive. In the velocity example, it tells us when the velocity stops increasing and starts to decrease, or vice-versa. In other words, when the acceleration changes from positive to negative.

This point is called an inflection point. In the f(x) graph, it indicates where the rate of change of angle of slope reverses.

Example

Mathematics question

For the function $y=x^5-5x^4$

$f'(x) = 5x^4-20x^3$

$f″(x) = 20x^3-60x^2$

Maxima and minima occur when $f'(x) = 0$ ⇒ $x=0$ and $x=4$

At $x=0$, $f″(x)=0$, but $f″(x)$ does not cross the $x$-axis at this point, so it is not an inflection point. When $x$ is a little less than $0$, $f'(x) > 0$, and when $x$ is a little greater than $0$, $f'(x) < 0$, so point $(0, 0)$ is a local maximum.

At $x=4$, $f″(x)>0$, but $f″(x)$ does not cross the $x$-axis at this point, so it is not an inflection point. When $x$ is a little less than $4$, $f'(x) < 0$, and when $x$ is a little greater than $0$, $f'(x) > 0$, so point $(4, -256)$ is a local minimum.

There is a horizontal inflection point at $(3, -162)$, where $f″(x)$ crosses the $x$-axis.

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