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[h] SL 5.8 Local maximum and minimum values

[q] **Turning/Inflexion Points**

As has been mentioned on a couple of occasions, a maximum or minimum point on a curve is the point which has a flat/horizontal tangent. In general, a flat line has a slope of 0. Therefore, if \(f'(x) = 0\), then \(x_1\) is a turning point, as in the diagram above.

Example 1: Find the turning point of \(f(x) = 3x^3 – 12x^2 + 4\):

\[

f'(x) = 6x – 12 = 0 \Rightarrow 6x = 12 \Rightarrow x = 2.

\]

Find \(y: f(2) = 3(2)^3 – 12(2)^2 + 4 = -8 \Rightarrow (2, -8)\).

[q]

TESTING FOR MAX OR MIN:

If we look at the minimum shown in this diagram, we see that the curve goes from a negative slope, to 0 (at the min), then positive. In other words, the slope is increasing around this point, which also means the rate of change of the gradient is positive, i.e., \(f”(x) > 0\).

– If we have a turning point at \(x_1\): If \(f”(x_1) > 0 \Rightarrow \) minimum

– Sometimes called the **2nd derivative test**: If \(f”(x_1) < 0 \Rightarrow \) maximum

[a]

OPTIMISATION:

One very useful application of these turning points is to optimise a real-world function, such as: minimising costs, maximising area, maximising profits, etc.

Example : A box has sides: \(x, 2x, 18 – 3x\). Find the \(x\) that minimises volume:

Find function for volume:

\[

V = x \cdot 2x \cdot (18 – 3x) = 36x^2 – 6x^3.

\]

To find \(x\):

\[

\frac{dV}{dx} = 72x – 18x^2 = 0 \Rightarrow 18x(4 – x) = 0.

\]

So, \(x = 0\) or \(x = 4\), but can’t have a side length of 0, so \(x = 4\).

[q]

INFLEXION POINTS:

In quadratics, you saw mentions of \(V\)-shaped & \(\land\)-shaped graphs. We will now see new terminology for these sections of curves:

We also saw on the previous page that a concave-up section of a curve is where \(f”(x) > 0\) and vice versa.

[a]

An inflexion (or inflection) point is simply where there is a change in concavity, from up to down, or down to up. If the second derivative is changing sign, then that is the point where we have:

NOTE: An inflection can also be a stationary point, i.e.:

NOTE 2: There are rare cases where you get \(f”(x) = 0\) but there is no change in concavity. This is not an inflection point; must meet both criteria.

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