Embibe Experts Solutions for Chapter: Continuity and Differentiability, Exercise 1: EXERCISE-1

If $f\left(x\right)=\frac{1-\tan x}{4x-\pi }, x\ne \frac{\pi }{4}, x\in \left[0,\frac{\pi }{2}\right)$ is a continuous function, then $f\left(\frac{\pi }{4}\right)$ is equal to

Let $f\left(x\right)=\frac{x\left(1+a\cos x\right)-b\sin x}{x^3}, x\ne 0$ and $f\left(0\right)=1$. The value of $a$ and $b$ so that $f$ is a continuous function are

'$f$' is a continuous function on the real line. Given that $x^2+\left(f\left(x\right)-2\right)x-\sqrt{3}·f\left(x\right)+2\sqrt{3}-3=0$. Then the value of $f\left(\sqrt{3}\right)$ is

The true set of real values of $x$ for which the function, $f\left(x\right)=x\left(\ln \left(x\right)\right)-x+1$ is positive is:

Rolle's theorem in the indicated intervals will not be valid for which of the following function

Consider the function for $x\in \left[-2,3\right], f\left(x\right)=\left\{\begin{array}{cc}\frac{x^3-2x^2-5x+6}{x-1},& x\ne 1\\ -6,& x=1\end{array}\right\}$, then

If the function $f\left(x\right)=2x^2+3x+5$ satisfies LMVT at $x=2$ on the closed interval $\left[1,a\right]$ then the value of $\text{'}a\text{'}$ is equal to

Consider the function $f\left(x\right)=8x^2-7x+5$ on the interval $\left[-6,6\right]$. The value of $c$ that satisfies the conclusion of the mean value theorem, is