Rolle's Theorem

Q.

If Rolle’s theorem holds for the function $f\left(x\right)=x^3-a{x}^2+bx-4,x\in [1,2]$ with $f^{}\left(\frac{4}{3}\right)=0$, then ordered pair $\left(a,b\right)$ is equal to

1. $(-5,8)$

2. $(5,8)$

3. $(5,-8)$

4. $(-5,-8)$

Q.

The equation of one of the lines represented by the equation $pq(x^2-y^2)+(p^2-q^2)xy=0$, is

1. $px+qy=0$

2. $px–qy=0$

3. $p^{2}x+q^{2}y=0$

4. $q^{2}x–p^{2}y=0$

Q.

If the function $f\left(x\right)=x^3-6x^2+ax+b$ satisfies Rolles theorem in the interval $\left[1,3\right]$ and $f\left[\frac{\left(2\sqrt{3}+1\right)}{\sqrt{3}}\right]=0$, then:

1. $a=-11$

2. $a=-6$

3. $a=6$

4. $a=11$

Q.

If $p$ and $q$ are prime numbers satisfying the condition $p^2-2q^2=1$, then the value of $p^2+2q^2$ is

1. $5$

2. $15$

3. $16$

4. $17$

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

1. 1
2. -1
3. 2
4. -3

Q. If $f$ and $g$ are differentiable functions in $[0,1]$ satisfying $f\left(0\right)=2=g\left(1\right),g\left(0\right)=0$ and $f\left(1\right)=6$, then for some $c\in [0,1]$:

1. $2f^{\prime }\left(c\right)=g^{\prime }\left(c\right)$
2. $2f^{\prime }\left(c\right)=3g^{\prime }\left(c\right)$
3. $f^{\prime }\left(c\right)=g^{\prime }\left(c\right)$
4. $f^{\prime }\left(c\right)=2g^{\prime }\left(c\right)$

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

1. $1$
2. $-1$
3. $2$
4. $-3$

Q. If $f:R\to R$ is a twice differentiable function such that $f^{\prime \prime }\left(x\right)>0$ for all $x\in R$, and $f\left(\frac{1}{2}\right)=\frac{1}{2},f\left(1\right)=1,\text{then}$

1. $f^{\prime }\left(1\right)\le 0$
2. $0<f^{\prime }\left(1\right)\le \frac{1}{2}$
3. $\frac{1}{2}<f^{\prime }\left(1\right)\le 1$
4. $f^{\prime }\left(1\right)>1$

Q.

Between any two real roots of the equation $e^x$ sin x = 1, the equation $e^x$ cos x = - 1 has

1. Atleast one root

2. Exactly one root

3. Atmost one root

4. No root

Q. If $c$ is a point at which Rolle's theorem holds for the function, $f\left(x\right)={\log }_e\left(\frac{x^2+\alpha }{7x}\right)$ in the interval $[3,4]$, where $\alpha \in R$, then $f^{\prime \prime }\left(c\right)$ is equal to:

1. $\frac{1}{12}$
2. $-\frac{1}{12}$
3. $-\frac{1}{24}$
4. $\frac{\sqrt{3}}{7}$

Q. If family of straight lines $ax+by+c=0$ always passes through a fixed point $(\frac{3}{2},1),$ then equation $36a{x}^2+8bx+2c=0$ has

1. at least one root in $[0,1]$
2. atleast one root in $[\frac{-1}{2},\frac{1}{2}]$
3. atleast one root in $[-1,2]$
4. atleast one root in $[0,\frac{1}{2}]$

Q.

According to Rolle’s theorem, there will be at least one solution for f’(x) = 0 in the interval [-2, 2], where $f\left(x\right)=\frac{1}{x^2}$

1. True

2. False