JEE Mains · Maths · STD 12 - 6. Application of derivatives
Let \( (2\alpha, \alpha) \) be the largest interval in which the function \( f(t)=\frac{|t+1|}{t^{2}}, t<0 \), is strictly decreasing. Then the local maximum value of the function \( g(x)=2\log_{e}(x-2)+\alpha x^{2}+4x-\alpha, x>2 \), is
- A 2
- B 3
- C 4
- D 5
Answer & Solution
Correct Answer
(C) 4
Step-by-step Solution
Detailed explanation
Drawing graph of f(t) for t < 0 \(g(x)=\log _e(x-2)-x^2+4 x+1 ; x>2\) \(g ^{\prime}( x )=\frac{2}{ x -2}-(2( x -2) ; x >2\) \(g ^{\prime}( x )=\frac{1-( x -2)^2}{( x -2)}=\frac{-( x -3)( x -1)}{( x -2)}\) as x > 2 maxima occur at x = 3 \(g(3)=2 \log _e 1-9+12+1=4\)
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