How do you find the turning points of a cubic function?
Use the first derivative test.
Given: How do you find the turning points of a cubic function?
The definition of A turning point that I will use is a point at which the derivative changes sign. According to this definition, turning points are relative maximums or relative minimums.
Use the first derivative test:
Then set up intervals that include these critical values.
If you also include turning points as horizontal inflection points, you have two ways to find them:
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To find the turning points of a cubic function, follow these steps:

Start with the cubic function in the form ( f(x) = ax^3 + bx^2 + cx + d ).

Calculate the first derivative of the function, denoted as ( f'(x) ), using differentiation rules.

Set ( f'(x) = 0 ) and solve for ( x ). These solutions give the critical points of the function.

Substitute the critical points into the original cubic function to find the corresponding ( y ) values.

The critical points where the sign of the second derivative changes correspond to turning points. To determine this, calculate the second derivative ( f''(x) ) and evaluate its sign at each critical point.

If ( f''(x) > 0 ), the turning point is a local minimum. If ( f''(x) < 0 ), the turning point is a local maximum.

The coordinates of the turning points are the ( (x, y) ) values obtained from substituting the critical points into the original cubic function.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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