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JEE Main202429 Jan 2024Evening ShiftMathematicsApplication of DerivativesActual

The function f x = x x 2 - 6 x - 16 , x ∈ ℝ - - 2 , 8

Options

  1. Adecreases in ( - 2 , 8 ) and increases in ( - ∞ , - 2 ) ∪ ( 8 , ∞ )
  2. Bdecreases in ( - ∞ , - 2 ) ∪ ( - 2 , 8 ) ∪ ( 8 , ∞ )
  3. Cdecreases in ( - ∞ , - 2 ) and increases in ( 8 , ∞ )
  4. Dincreases in ( - ∞ , - 2 ) ∪ ( - 2 , 8 ) ∪ ( 8 , ∞ )

Correct answer

B. decreases in ( - ∞ , - 2 ) ∪ ( - 2 , 8 ) ∪ ( 8 , ∞ )

Step-by-step solution

Given: f x = x x 2 - 6 x - 16 Now differentiating the above function we get, ⇒ f ' x = x 2 - 6 x - 16 · 1 - x · 2 x - 6 x 2 - 6 x - 16 2 ⇒ f ' x = - x 2 - 16 x 2 - 6 x - 16 2 ⇒ f ' x < 0 . . . i Now, x 2 - 6 x - 16 ≠ 0 ⇒ x 2 - 8 x + 2 x - 16 ≠ 0 ⇒ x + 2 x - 8 ≠ 0 ⇒ x ≠ 8 , - 2 . . . i i Using i and i i , So, the function is always decreasing in x ∈ ( - ∞ , - 2 ) ∪ ( - 2 , 8 ) ∪ ( 8 , ∞ ) .

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