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GATE 2017 EE – Question 36

Engineering Mathematics · Calculus: Mean value theorems, Theorems of integral calculus, Evaluation of definite and improper integrals, Partial derivatives, Multiple integrals, Fourier series · 2 marks · Multiple choice

A function $f(x)$ is defined as $f(x) = \begin{cases} e^x, & x < 1 \\ \ln x + ax^2 + bx, & x \geq 1 \end{cases}$, where $x \in \mathbb{R}$. Which one of the following statements is TRUE?

  1. $f(x)$ is NOT differentiable at $x = 1$ for any values of $a$ and $b$.
  2. $f(x)$ is differentiable at $x = 1$ for the unique values of $a$ and $b$.
  3. $f(x)$ is differentiable at $x = 1$ for all values of $a$ and $b$ such that $a + b = e$.
  4. $f(x)$ is differentiable at $x = 1$ for all values of $a$ and $b$.

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Correct answer: (B) $f(x)$ is differentiable at $x = 1$ for the unique values of $a$ and $b$.

Explanation

For $f$ to be differentiable at 1 it must first be continuous, so $e = a + b$. The derivatives must also match: from the left $e$, and from the right $1 + 2a + b$. So $2a + b = e - 1$. Solving the two equations gives $a = -1$ and $b = e + 1$, which is a unique pair of values.