Calculus question

Question

Excuse the black blob

 

Answer 1

The function f(x) = 5 + 24x - 2x³

f '(x) = - 6x² + 24

Equate f '(x) to zero, to get critical points.

- 6x² + 24 = 0

- 6x² = - 24

x² = 4

x = ± 2

If f '(x) > 0 (Positive) for all x in (a, b), then f(x) is increasing on [a, b].

If f '(x) < 0 (Negative) for all x in (a, b), then f(x) is decreasing on [a, b].

Test intervals    x - Value          Polynomial or f '(x) value             Conclusion

(- ∞, - 2)           x = - 3         - 6(-3)² + 24 = -54+24 = - 30 < 0      Decreasing

(- 2, 2)              x = 1           - 6(1)² + 24 = - 6+24 = 18 > 0           Increasing

(2, ∞)                x = 3           - 6(3)² + 24 = -54+24 = -30 < 0        Decreasing

So, f(x) is increasing on the interval (- 2, 2)

f '(x) = - 6x² + 24

f ''(x) =  - 12x

- 12x = 0

x = 0

x = 0 is the point of inflection.

Substitute x = 0 in f(x) = 5 + 24x - 2x³

y = 5 + 24(0) - 2(0)³

y = 5

(0,5) point of inflection.

If f " (x) > 0 (Positive) for all x in (a, b), then f(x) is concave upward on (a, b).

If f " (x) < 0 (Negative) for all x in (a, b), then f(x) is concave downward on (a, b).

To find the Concavity, locate the x - values at which f "(x) = 0 or f " (x) does not exist.

Test intervals       x - Value                Polynomial or f " (x) value       Conclusion

(- ∞,0)                  x = - 1                  - 12(-1) = 12       > 0              Concave upward.

(0, ∞)                   x = 1                   - 12(1) = -12 = - 12 < 0            Concave downward.   

  Concave downward   in the interval   0 < x < ∞

 c = 0

Comments

Apparently for the first one, one of the answers is infinity? So i am very confused

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Modified the solution please check.

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Is it not (- ∞,0) where it is increasing?

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It says one answer is infinity but that would only work for the derivative?

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Since the critical number - 2 lies in the interval (- ∞, 0). So the function change (decreasing or increasing) at x = - 2.

The function f(x) is decreasing in the interval (- ∞, - 2) and the function f(x) is increasing in the interval (-2, 2) and observe the graph below.

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Thank you, i asked my teacher and the answer in the textbook was wrong :) this is correct.