If $a+b+c=0$ then the equation $3a{x}^2+2bx+c=0$ has:

Important Questions on Continuity and Differentiability

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Achieve CUET (UG) Mathematics Practice Book>Chapter 5 - Continuity and Differentiability>Exercise>Q 110

If Rolle's theorem holds for the function $f\left(x\right)=2x^3+b{x}^2+cx, x\in \left[-1,1\right]$ at the point $x=\frac{1}{2},$ then $2b+c$ is equal to

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Achieve CUET (UG) Mathematics Practice Book>Chapter 5 - Continuity and Differentiability>Exercise>Q 111

Let $f$ be the differentiable for $\forall$ $x$. If $f\left(1\right)=-2$ and $f^{\text{'}}\left(x\right)\ge 2$   for $[1,6]$, then

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Achieve CUET (UG) Mathematics Practice Book>Chapter 5 - Continuity and Differentiability>Exercise>Q 112

In the mean value theorem $\frac{f\left(b\right)-f\left(a\right)}{b-a}=f^{\prime }\left(c\right)$, if $a=0, b=\frac{1}{2}$ and $f\left(x\right)=x\left(x-1\right)$$(x-2)$, then value of $c$ is -

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Achieve CUET (UG) Mathematics Practice Book>Chapter 5 - Continuity and Differentiability>Exercise>Q 113

If $3\left(a+2c\right)=4\left(b+3d\right)\ne 0,$ then the equation $a{x}^3+b{x}^2+cx+d=0$ will have

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Achieve CUET (UG) Mathematics Practice Book>Chapter 5 - Continuity and Differentiability>Exercise>Q 114

Let $f\left(x\right)$ be a differentiable function in the interval $\left(0,2\right)$, then the value of ${\int }_0^{2}f\left(x\right)dx$

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Achieve CUET (UG) Mathematics Practice Book>Chapter 5 - Continuity and Differentiability>Exercise>Q 115

In which of the following functions is Rolle’s theorem applicable?

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Achieve CUET (UG) Mathematics Practice Book>Chapter 5 - Continuity and Differentiability>Exercise>Q 116

Given, $f\left(x\right)=4-{\left(\frac{1}{2}-x\right)}^{\frac{2}{3}}, g\left(x\right)=\left\{\begin{array}{cc}\frac{\tan \left[x\right]}{x},& x\ne 0\\ 1& x=0\end{array}\right.$$, h\left(x\right)=\left\{x\right\}, k\left(x\right)=5^{\mathrm{l}\mathrm{o}{\mathrm{g}}_2\left(x+3\right)},$ then in $\left[\mathrm{0,} 1\right]$, Lagrange’s mean value theorem is not applicable to, where $\left[x\right]$ and $\left\{x\right\}$ denote the greatest integer and fraction part function.

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Achieve CUET (UG) Mathematics Practice Book>Chapter 5 - Continuity and Differentiability>Exercise>Q 117

A value of $c$ for which conclusion of Mean Value Theorem holds for the function $f\left(x\right)={\log }_e\left(x\right)$ on the interval $\left[1, 3\right]$ is