## Turning Points

### In mathematics, turning points are the points where a curve changes its direction from increasing to decreasing or vice versa

In mathematics, turning points are the points where a curve changes its direction from increasing to decreasing or vice versa. They are often referred to as critical points, stationary points, or points of inflection. Turning points play an important role in analyzing and understanding the behavior of curves and functions.

To determine the turning points of a curve or function, we need to find the points where its derivative is zero or undefined. The derivative of a function describes its rate of change or its slope at any given point. At a turning point, the slope changes from positive to negative or vice versa.

Here is a step-by-step approach to finding turning points:

1. Start by finding the first derivative of the function. The first derivative represents the rate of change of the function with respect to the independent variable.

2. Set the first derivative equal to zero and solve for the independent variable. This will give us the x-coordinates of the turning points.

3. Substitute the x-coordinates obtained in step 2 back into the original function to find the corresponding y-coordinates.

4. Finally, we have the turning points as coordinates (x, y).

Let’s illustrate this process with an example:

Example: Find the turning points of the function f(x) = x^3 – 6x^2 + 9x + 1.

Step 1: Find the first derivative of f(x).

f'(x) = 3x^2 – 12x + 9

Step 2: Set f'(x) = 0 and solve for x.

3x^2 – 12x + 9 = 0

(x – 1)(3x – 3) = 0

This gives us two possible x-coordinates: x = 1 and x = 1/3.

Step 3: Substitute the x-coordinates back into the original function.

For x = 1, f(1) = (1)^3 – 6(1)^2 + 9(1) + 1 = 5.

For x = 1/3, f(1/3) = (1/3)^3 – 6(1/3)^2 + 9(1/3) + 1 = 8/27.

So, the first turning point is (1, 5) and the second turning point is (1/3, 8/27).

Remember that turning points can also be points of inflection, where the concavity of the curve changes, but the slope does not necessarily change. To find these points, you would need to examine the second derivative of the function.

It’s worth noting that not every function will have turning points. For example, a linear function does not have any turning points since its rate of change is constant. Turning points are more commonly found in polynomial or trigonometric functions.

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