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Increasing and Decreasing Functions — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsIncreasing and Decreasing Functions1 Mark
Question
If the function f(x) = kx3 - 9x2 + 9x + 3 is monotonically increasing in every interval, then:
  1. $\text{k}<3\text{k}<3$
  2. $\text{k}\leq3\text{k}\leq3$
  3. $\text{k}>3\text{k}>3$
  4. $\text{k}\geq3$
✓

Answer

  1. k > 3k > 3

Solution:

f(x) = kx3 - 9x2 + 9x + 3

f'(x) = kx2 - 27

= 3(x2 - 9)

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

f'(x) > 0

⇒ 3(x2 - 9) > 0

⇒ (x2 - 9) > 0 [Since, 3 > 0, 3(x2 - 9) > 0 ⇒ (x2 - 9) > 0]

⇒ (x + 3)(x - 3) > 0

⇒ x < -3 or x > 3

⇒ |x| > 3

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