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Increasing and Decreasing Functions — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIncreasing and Decreasing Functions1 Mark
Question
Function f(x) = 2x3 - 9x2 + 12x + 29 is monotonically decreasing when:
  1. x < 2
  2. x > 2
  3. x > 3
  4. 1 < x < 2
✓

Answer

  1. 1 < x < 2

Solution:

f(x) = 2x3 - 9x2 + 12x + 29

⇒ f'(x) = 6x2 - 18x + 12

⇒ f'(x) = 6(x2 - 3x + 2)

⇒ f'(x) = 6(x - 1)(x - 2)

For f(x) to be decreasing, we must have

 f'(x) < 0

⇒ 6(x - 1)(x - 2) < 0

⇒ (x - 1)(x - 2) < 0

[Since, 6 > 0, 6(x - 1)(x - 2) < 0 ⇒ (x - 1)(x - 2) < 0]

⇒ 1 < x < 2

So, f(x) is decreasing for 1 < x < 2.

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