Before You Begin

Question #41374

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to \mathbb{R}$ be defined as $f(x) = \int_0^x (\{t\}^3 - \{t\}) dt$ where $\{t\}$ denotes the fractional part of $t$ . Which of the following statements is CORRECT?

Options

A.
$f(x)$ has a local maximum at each integer $k$ , and $f''(k) = -1$ .
B.
$f(x)$ is strictly decreasing on $\mathbb{R}$ , and $\lim_{x \to k^-} f''(x) - \lim_{x \to k^+} f''(x) = 3$ for all $k \in \mathbb{Z}$ .
C.
$f(x)$ has a local minimum at each integer $k$ , and $f(x)$ is non-differentiable at $x=k$ .
D.
$f(x)$ is strictly decreasing on $\mathbb{R}$ , and $\lim_{x \to k^-} f''(x) + \lim_{x \to k^+} f''(x) = 3$ for all $k \in \mathbb{Z}$ .

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

Second Derivative Test Limitations|Leibniz Rule for Differentiation under Integral|Greatest Integer Function Traps

Why This Question Is Tricky

Common Mistakes

This question tests the limitations of the Second Derivative Test and the differentiability of integrals involving the fractional part function.

A common trap is mechanically applying the formula $f''(x) = 3\{x\}^2 - 1$ and evaluating it at integer points (yielding $-1$ ), leading students to falsely conclude that $x=k$ are local maxima.

However, because the integrand's derivative has a jump discontinuity at integers, $f''(x)$ is undefined there.

Furthermore, a simple first-derivative sign analysis reveals that $f'(x) \le 0$ everywhere, meaning the function is strictly decreasing and the critical points are merely points of inflection, not local extrema.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	Second Derivative Test Limitations
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free