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Local extrema and increasing or decreasing intervals of a cubic

The rule: critical points split the number line into sign intervals

Differentiate, solve $f^{'} (x) = 0$ , and test the sign of $f^{'}$ on each resulting interval. Positive derivative means increasing; negative means decreasing. A change from positive to negative gives a local maximum, and a change from negative to positive gives a local minimum.

Problem

For $f (x) = x^{3} - 4 x + 5$ on $[- 2, 2]$ , find the local maximum and local minimum values. Determine where the function is increasing and where it is decreasing. Round decimal answers to two places.

Answer

[- 2, 2]

, the local maximum is about

8.08

x \approx - 1.15

, and the local minimum is about

1.92

x \approx 1.15

. The function increases on

[- 2, - 1.15] \cup [1.15, 2]

and decreases on

(- 1.15, 1.15)

The rounded graphing answers were verified from the exact derivative and exact critical points.

Step-by-step solution

Find the critical points

Differentiate:

f^{'} (x) = 3 x^{2} - 4

Set the derivative equal to zero:

3 x^{2} - 4 = 0

x^{2} = \frac{4}{3}

x = \pm \frac{2}{3} \approx \pm 1.15

Build the sign chart

For $x < - 2/ 3$ , the term $3 x^{2} - 4$ is positive, so $f$ is increasing. Between the two critical points it is negative, so $f$ is decreasing. After $2/ 3$ it is positive again, so $f$ is increasing. Restricted to $[- 2, 2]$ , this gives

increasing on [- 2, - 1.15] \cup [1.15, 2]

decreasing on (- 1.15, 1.15)

Evaluate and classify the turning points

At the negative critical point, the derivative changes from positive to negative, so this is a local maximum:

f (- \frac{2}{3}) = 5 + \frac{16}{3 3} \approx 8.08

At the positive critical point, the derivative changes from negative to positive, so this is a local minimum:

f (\frac{2}{3}) = 5 - \frac{16}{3 3} \approx 1.92

How to check yourself

The cubic has a positive leading coefficient, so its direction pattern should be up, down, up. Also $f (0) = 5$ , which lies between the two turning values and confirms that the middle branch descends from about $8.08$ to about $1.92$ .

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