Before You Begin

Question #41398

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x) = \begin{cases} x^3 + ax^2 + bx + c, & x < 1 \\ \max\{2 - |x-2|, x^2 - 4x + 4\}, & x \ge 1 \end{cases}$ where $a, b, c \in \mathbb{R}$ . If $f(x)$ is differentiable at $x = 1$ and has a local extremum at $x = -1$ , then the sum of all values of $x$ where $f(x)$ attains a local extremum is:

Options

A.
$\frac{29}{6}$
B.
$\frac{35}{6}$
C.
4
D.
$-\frac{1}{6}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Non-Differentiable Extrema|Polynomial Continuity/Differentiability Link|Limit Existence at Discontinuity

Why This Question Is Tricky

Common Mistakes

This question deploys the 'Lethal Collision' architecture by splicing a polynomial with a non-differentiable piecewise function.

The primary trap tests the boundary of the 'Critical Point' definition.

Students often incorrectly assume that local extrema only occur where $f'(x) = 0$ , completely missing the points where $f'(x)$ does not exist (such as $x=2$ and $x=3$ generated by the absolute value and max functions).

Furthermore, the junction point $x=1$ acts as a decoy; students frequently flag piecewise boundaries as extrema by default, but the forced differentiability constraint ( $f'(1)=1$ ) ensures the function is strictly increasing there, disqualifying it.

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