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NDA 2024 Maths · Mathematics · Application of Derivatives · medium

The function f(x)=x^3-6x^2+9x+2 is strictly increasing in

Single correct option

The function f(x)=x3−6x2+9x+2f(x)=x^3-6x^2+9x+2 is strictly increasing in
  1. A. (1,3)(1,3)
  2. B. (−∞,3)(-\infty,3)
  3. C. (−∞,1)∪(3,∞)(-\infty,1)\cup(3,\infty)
  4. D. (1,∞)(1,\infty)
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Answer: (C) (−∞,1)∪(3,∞)(-\infty,1)\cup(3,\infty)

f′(x)=3x2−12x+9=3(x−1)(x−3)f'(x)=3x^2-12x+9=3(x-1)(x-3), which is positive for x<1x<1 and for x>3x>3 and negative on (1,3)(1,3). Hence ff increases on (−∞,1)∪(3,∞)(-\infty,1)\cup(3,\infty). Trap: (1,3)(1,3) is where ff decreases, and the other options ignore one of the sign changes.
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