Before You Begin

Question #41400

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = |x^5 - 7x^4 + 18x^3 - 22x^2 + 13x - 3|$ . Let $M$ be the sum of the abscissae of all points of local minimum of $f$ , and $N$ be the sum of the abscissae of all points of local maximum of $f$ . Find the value of $10M + 5N$ .

Options

A.
53
B.
43
C.
23
D.
48

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Non-Differentiable Extrema|Critical Point vs Stationary Point|Even Multiplicity Roots

Why This Question Is Tricky

Common Mistakes

This question targets the mechanical application of the first and second derivative tests.

By placing a polynomial with an even multiplicity root ( $x=1$ ) and a single root ( $x=3$ ) inside a modulus, it creates two distinct types of minima.

At $x=1$ , the function is differentiable, but the second derivative test fails ( $f''(1)=0$ ), trapping students who assume it must be an inflection point.

At $x=3$ , the function is non-differentiable, trapping students who only look for stationary points ( $f'(x)=0$ ).

Successfully solving this requires analyzing the sign change of the first derivative and recognizing non-differentiable critical points.

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