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# Identifying Turning Points (Local Extrema) for a Function

## Key Questions

See below.

#### Explanation:

To find extreme values of a function $f$, set $f ' \left(x\right) = 0$ and solve. This gives you the x-coordinates of the extreme values/ local maxs and mins.

For example. consider $f \left(x\right) = {x}^{2} - 6 x + 5$. To find the minimum value of $f$ (we know it's minimum because the parabola opens upward), we set $f ' \left(x\right) = 2 x - 6 = 0$ Solving, we get $x = 3$ is the location of the minimum. To find the y-coordinate, we find $f \left(3\right) = - 4$. Therefore, the extreme minimum of $f$ occurs at the point $\left(3 , - 4\right)$.

• Any polynomial of degree $n$ can have a minimum of zero turning points and a maximum of $n - 1$. However, this depends on the kind of turning point.

Sometimes, "turning point" is defined as "local maximum or minimum only". In this case:

• Polynomials of odd degree have an even number of turning points, with a minimum of 0 and a maximum of $n - 1$.
• Polynomials of even degree have an odd number of turning points, with a minimum of 1 and a maximum of $n - 1$.

However, sometimes "turning point" can have its definition expanded to include "stationary points of inflexion". For an example of a stationary point of inflexion, look at the graph of $y = {x}^{3}$ - you'll note that at $x = 0$ the graph changes from convex to concave, and the derivative at $x = 0$ is also 0.

If we go by the second definition, we need to change our rules slightly and say that:

• Polynomials of degree 1 have no turning points.
• Polynomials of odd degree (except for $n = 1$) have a minimum of 1 turning point and a maximum of $n - 1$.
• Polynomials of even degree have a minimum of 1 turning point and a maximum of $n - 1$.

So, in part, it depends on the definition of "turning point", but in general most people will go by the first definition.

• For a differentiable function $f \left(x\right)$, at its turning points, $f '$ becomes zero, and $f '$ changes its sign before and after the turning points.

I hope that this was helpful.