Exams › JEE Main › Maths

Select the statement(s) that are true: (a) If the derivative f'(a) exists and is finite at a point, then f must be continuous at x = a. (b) The function f(x) = 3\tan(5x) - 7 is differentiable at every point where it is defined.

1. (a) only
2. (b) only
3. Both (a) and (b)
4. Neither (a) nor (b)

Correct answer: Neither (a) nor (b)

Solution

Statement (a) is false because a function can have a derivative at a point without being continuous there, such as in cases of removable discontinuities. Statement (b) is also false because the tangent function has vertical asymptotes where it is undefined, making the overall function non-differentiable at those points.

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 →