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

The function fx=xx2-6x-16,x∈ℝ--2,8
  1.  decreases in (-2,8) and increases in (-∞,-2)∪(8,∞)
  2. decreases in (-∞,-2)∪(-2,8)∪(8,∞)
  3. decreases in (-∞,-2) and increases in (8,∞)
  4. increases in (-∞,-2)∪(-2,8)∪(8,∞)

Solution

Given: fx=xx2-6x-16

Now differentiating the above function we get,

⇒f'x=x2-6x-16·1-x·2x-6x2-6x-162

⇒f'x=-x2-16x2-6x-162

⇒f'x<0   ...i

Now, x2-6x-16≠0

⇒x2-8x+2x-16≠0

⇒x+2x-8≠0

⇒x≠8,-2   ...ii

Using i and ii,

So, the function is always decreasing inx∈(-∞,-2)∪(-2,8)∪(8,∞).

Asked in: JEE Main 2024 (29 Jan Shift 2)

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