How do you find the turning points of a cubic function?

Answer 1

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:

First find the first derivative #f'(x)#
Set the #f'(x) = 0# to find the critical values.

Then set up intervals that include these critical values.

Select test values of #x# that are in each interval.
Find out if #f'#(test value #x#) #< 0# or negative
Find out if #f'#(test value #x#) #> 0# or positive.
A relative Maximum: #f'("test value "x) >0, f'("critical value") = 0, f'("test value "x) < 0#
A relative Minimum: #f'("test value "x) <0, f'("critical value") = 0, f'("test value "x) > 0#

If you also include turning points as horizontal inflection points, you have two ways to find them:

#f'("test value "x) >0, f'("critical value") = 0, f'("test value "x) > 0#
#f'("test value "x) <0, f'("critical value") = 0, f'("test value "x) < 0#
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Answer 2

To find the turning points of a cubic function, follow these steps:

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

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

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

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

  5. 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.

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

  7. 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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Answer from HIX Tutor

When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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