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Question #41326

Difficulty: 8/10 | 🟢 Free

The Question

Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = \int_0^x (t^2 - 2at + 3a + 4) e^{t^2} dt$ where $a$ is a real parameter. Let $S$ be the set of all integers $a$ such that $f(x)$ is invertible and its inverse function $g(x)$ has a continuous second derivative for all $x$ in the domain of $g$ . Find the sum of all elements in $S$ .

Options

A.
4
B.
6
C.
9
D.
10

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

Second Order Derivative Continuity|Inverse Function Integration|Bijectivity and Inverse

Why This Question Is Tricky

Common Mistakes

This question tests the subtle distinction between a function being invertible (bijective) and its inverse having a continuous second derivative.

Students often correctly identify that for $f(x)$ to be invertible, $f'(x) \ge 0$ is sufficient, leading to the closed interval $a \in [-1, 4]$ .

However, they blindly apply the formula $g''(y) = -f''(x)/[f'(x)]^3$ without realizing that if $f'(x) = 0$ at any point (which happens at the boundaries $a = -1$ and $a = 4$ ), the second derivative of the inverse blows up to infinity.

The lethal collision occurs when students fail to transition from the global property of bijectivity to the local requirement of non-zero derivatives for higher-order smoothness of the inverse, trapping them into answering 9 instead of 6.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Continuity and Differentiability - Class 12
Concept	Second Order Derivative Continuity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free