Exams › JEE Main › Maths
Consider the function f: R → R given by f(x) = 5, for x ≤ 1; a + bx, for 1 < x < 3; b + 5x, for 3 ≤ x < 5; 30, for x ≥ 5. Which of the following is true about f?
- f is continuous when a = 5 and b = 5
- f is continuous when a = −5 and b = 10
- f is continuous when a = 0 and b = 5
- f is never continuous, whatever the values of a and b
Correct answer: f is continuous when a = 0 and b = 5
Solution
The function is continuous at the transition points when the left-hand limit, right-hand limit, and the function value at those points are equal. Setting a = 0 and b = 5 ensures that the values match at x = 1 and x = 3, thus maintaining continuity throughout the defined intervals.
Related JEE Main Maths questions
- Suppose $f(x)=\alpha(x)\beta(x)\gamma(x)$ for every real $x$, where $\alpha(x)$, $\beta(x)$, and $\gamma(x)$ are differentiable. If $f'(2)=18f(2)$, $\alpha'(2)=3\alpha(2)$, $\beta'(2)=-4\beta(2)$, and $\gamma'(2)=k\gamma(2)$, then what is the value of $k$?
- For the piecewise-defined function $$f(x)=\begin{cases} \dfrac{\sin((a+1)x)+\sin x}{x}, & x<0,\\[4pt] c, & x=0,\\[4pt] \dfrac{\sqrt{x+bx^2}-\sqrt{x}}{b x^{3/2}}, & x>0, \end{cases}$$ to be continuous at $x=0$, which values of $a$, $b$, and $c$ are required?
- Let $f(x)$ and $g(x)$ be continuously differentiable functions such that $f'(x)=g(x)$ and $f''(x)=-f(x)$. Define $h(x)=[f(x)]^2+[g(x)]^2$. If $h(0)=5$, what is the value of $h(10)$?
- For what value of $p$ can the function \[ f(x)=\begin{cases} \dfrac{(4^x-1)^3}{\sin x\,[\log|1+x^2/3|]^p}, & x\ne 0,\\[6pt] 12(\log 4)^3, & x=0, \end{cases} \] be made continuous at $x=0$?
- Using the mean value theorem, \(\dfrac{f(b)-f(a)}{b-a}=f'(c)\), take $a=0$, $b=\frac12$, and $f(x)=x(x-1)(x-2)$. What is the value of $c$?
- Consider the function $f:\mathbb{R}\to\mathbb{R}$ given by $f(x)=\max\{x,x^3\}$. At which points is $f(x)$ not differentiable?
⚔️ Practice JEE Main Maths free + battle 1v1 →