Question

Let \( f: \mathbb{R} \to \mathbb{R} \) be a twice differentiable function such that \( f''(x)>0 \) for all \( x \in \mathbb{R} \) and \( f'(a-1) = 0 \), where \( a \) is a real number. Let \( g(x) = f(\tan^2 x - 2\tan x + a) \), \( 0<x<\frac{\pi}{2} \).
Consider the following two statements :
(I) \( g \) is increasing in \( (0, \frac{\pi}{4}) \)
(II) \( g \) is decreasing in \( (\frac{\pi}{4}, \frac{\pi}{2}) \)
Then,

• A. Only (II) is True
• B. Only (I) is True
• C. Both (I) and (II) are True
• D. Neither (I) nor (II) is True

Hint:

If \( f''(x)>0 \), then \( f'(x) \) is strictly increasing. Use the point where the derivative is zero to establish the sign of \( f' \) in other regions.

Answer 1

The Correct Option is D

To solve the given problem, we need to analyze the behavior of the function \( g(x) = f(\tan^2 x - 2\tan x + a) \) and its properties.

1. \(f\colon \mathbb{R} \to \mathbb{R}\) is a twice differentiable function with \(f''(x) > 0\) for all \(x \in \mathbb{R}\). This implies that \(f\) is a convex function on the real line.
2. We are given that \(f'(a-1) = 0\). For a convex function, the derivative being zero at any point indicates it is a local minimum, as there are no maxima in a convex function.
3. Define \(h(x) = \tan^2 x - 2\tan x + a\) and note that the derivative \(h'(x) = 2\tan x \sec^2 x - 2\sec^2 x = 2 \sec^2 x (\tan x - 1)\).
   • In the interval \((0, \frac{\pi}{4})\), \(\tan x < 1\), hence \(h'(x) < 0\). This means \(h(x)\) is a decreasing function.
   • In the interval tan x > 1, hence \(h'(x) > 0\). This means \(h(x)\) is an increasing function.
4. Considering \(g(x) = f(h(x))\):
   • On \((0, \frac{\pi}{4})\), \(h(x)\) decreases while \(f\) being convex, decreases as we move away from its local minimum (where \(f'\) changes from zero). Consequently, \(g(x)\) is decreasing.
   • On h(x) increases, thus \(f(x)\) increases as well since we are moving horizontally away from the point where \(f'(a-1) = 0\) on either side. Thus, \(g(x)\) is increasing.

Therefore, neither of the assertions matches as expected:

• (I) "g is increasing in \( (0, \frac{\pi}{4}) \)" is false because \(g(x)\) actually decreases.
• (II) "g is decreasing in \( (\frac{\pi}{4}, \frac{\pi}{2}) \)" is false because \(g(x)\) actually increases.

Ultimately, the correct answer is: Neither (I) nor (II) is True.