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Question #41383

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to \mathbb{R}$ be a function defined by $f(x) = \begin{cases} \frac{1 + (x-1)^2 + \frac{(x-1)^4}{2} - e^{(x-1)^2}}{(x-1)^2} & \text{if } x < 1 \\ x^5 - 6x^4 + 14x^3 - 16x^2 + 9x - 2 & \text{if } x \ge 1 \end{cases}$ Let $S$ be the set of all points of inflection of the curve $y = f(x)$ . Find the sum of the elements of $S$ .

Options

A.
$\frac{8}{5}$
B.
$\frac{13}{5}$
C.
$\frac{3}{5}$
D.
1

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

Point of Inflection Conditions|Polynomial Continuity/Differentiability Link|Closed vs Open Intervals|L'Hopital's Pitfall

Why This Question Is Tricky

Common Mistakes

This question tests the rigorous definition of a point of inflection, specifically the necessity of a sign change in the second derivative.

The trap is set by a piecewise function that is twice differentiable at the junction $x=1$ with $f''(1)=0$ .

Students often mistakenly assume $f''(c)=0$ is sufficient for an inflection point and will incorrectly include $x=1$ in their set $S$ , leading to the trap answer 13/5.

The problem requires careful analysis of the sign of $f''(x)$ on both sides of the junction, utilizing Taylor series for the transcendental part to prove it remains strictly negative, thus revealing $x=1$ as a 'ghost' inflection point.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	Point of Inflection Conditions
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free