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Question #41399

Difficulty: 8/10 | 🟢 Free

The Question

Let $N(x_N, 0)$ be the point where the normal to the parabola $y^2 = 4x$ at the point $P(t^2, 2t)$ , for $t > 0$ , intersects the $x$ -axis. Let $f(t) = |x_N - 5|^{\frac{1}{t}}$ . Find the total number of local extrema of the function $f(t)$ for $t \in (0, \infty)$ .

Options

A.
1
B.
2
C.
3
D.
4

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Non-Differentiable Extrema|Normal to Conic|Logarithmic Differentiation Domain

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of logarithmic differentiation domains and non-differentiable extrema.

Students often blindly apply $f'(x)=0$ without considering the absolute value required to satisfy the domain of the logarithmic function.

By doing so, they miss the non-differentiable cusp at $t=\sqrt{3}$ , which acts as a local minimum.

Furthermore, analyzing the derivative requires splitting the domain and proving the monotonicity of the resulting transcendental equations, trapping those who rely solely on algebraic roots.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	Non-Differentiable Extrema
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free