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## Finding the turning points of a polynomial

The turning points are sometimes called the stationary points or maximums and minimums.

Lets look at what happens to the gradient at a turning point

- If we consider the maximum, you can see that before it has a positive gradient and after it has a negative gradient. During this transition it has a gradient of zero. So at a maximum
- If we consider the minimum, you can see that before it has a negative gradient and after it has a positive gradient. During this transition it has a gradient of zero. So at a minimum
- So we can see that at a turning point (maximum or minimum)

Examples

### Find the maximum of the curve

- We need to find the gradient function
- We let , because this is when the maximum will occur
- We solve the equation to find the value of x where the maximum occurs
- We substitute this value in the original equation to get the value for y

Let

If

the maximum value is

### Find the coordinates of the turning points of

- We need to find the gradient function
- We let , because this is when the turning points will occur
- We solve the equation to find the values of x where the turning points occurs
- We substitute the values into the original equation to get the values for y

Let

So of

If then and if then

So the coordinates of the turning points are and

### Given that the curve has a stationary point at find the value of a

- We need to find the gradient function
- We let , because this is when the turning points will occur and we will let
- We will solve the resulting equation to find the value of a

when

### Find the equation of the tangent to the curve at

- We need to find the gradient function , because to get the equation of a line we need the gradient and a point it passes through
- Substitute into the equation for y to get the required point the line passes through
- Substitute into the equation for to get the gradient of the line
- Use to get the equation of the line

At

At

So substituting into gives

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Categories: Differentiation

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