Skip to main content
Yukti Vault

Differentiation and its Application - JEE Mathematics question (JEE Advanced 2007 Paper 2)

Comprehension · Questions 1–1

Read the passage given below and answer the questions that follow.

If a continuous function $f$ defined on the real line $\mathbf R$, assumes positive and negative values in $\mathbf R$ then the equation $f(x)=0$ has a root in $\mathbf R$. For example, if it is known that a continuous function $f$ on $\mathbf R$ is positive at some point and its minimum value is negative then the equation $f(x)=0$ has a root in $\mathbf R$. Consider $f(x)=ke^x-x$ for all real $x$ where $k$ is a real constant.
JEE Advanced 2007 Paper 2Question 1
The line $y=x$ meets $y=ke^x$ for $k\leq0$ at
Differentiation and its Application
Options for question 1, choose one

Solution

Concept. Here $f(x)=ke^x-x$; its roots are the meeting points of $y=ke^x$ and $y=x$. For $k\le0$, $f$ is strictly decreasing. Formulas used. (1) A strictly monotonic continuous function going from $+\infty$ to $-\infty$ has exactly one rootStep 1 - Derivative. $f'(x)=ke^x-1<0$ for $k\le0$. Step 2 - Limits. $f\to+\infty$ as $x\to-\infty$ (the $-x$ term) and $f\to-\infty$ as $x\to\infty$. Step 3 - Conclude. By (1), exactly one root: the line meets the curve at one point. Answer: (B)

More from Differentiation and its Application