Topics:
How do you calculate MVT?
Two procedure mentioned here
Q1.Description of Rolle's Theorem :
All possible questions:
QA.How do you do Rolle's theorem?
QB.What are the three conditions of Rolle's theorem?
QC.What is the conclusion of Rolle's theorem?
QD.How do you find C in Rolle's theorem?
1.Rolle's Theorem
a.$\small f(x)$ is continuous at $\small [a,b]$
b.$\small f(x)$ is differentiable at $\small (a,b)$
c.$\small f(a)=f(b)$, there is $\small a<c<b$ for which $\small f^{'}(c)=0$.
Q2.Description of Lagranges Mean Value Theorem:
QA.What is LMV Theorem?
QB.How do you verify Lagrange's value theorem?
2.Lagranges Mean Value Theorem
All possible questions:
a.$\small f(x)$ is continuous at $\small [a,b]$
b.$\small f(x)$ is differentiable at $\small (a,b)$
c.$\small f(a)=f(b)$, there is $\small a<c<b$ for which $\small f^{'}(c)=\frac{f(b)-f(a)}{b-a}$.
Q3.If $\small a+b+c=0$, then the equation $\small 3ax^2+2bx+c=0$ has
a.Exist only one solution
b.There exist no solution
c.There exist maximum one solution
d.No Solution
Ans:b
Q4.If $\small a_{n}x^n+a_{n-1}x^{n-1}+....+a_{1}$ equation has a positive solution say,$\small \alpha$,then the solution is
a.$\small < \alpha$
b.$\small > \alpha$
c.$\small =\alpha$
d.$\small \geq \alpha$
Ans:a
Q5.$\small xlogx=3-x$ this equation has _______________ number of solution in this interval of (1,3)
a.Only one solution
b.Exactly one solution
c.No Solution
d,.Maximum one solution
Ans:a
Q6.$\small \phi(x)=a^{sinx}$ for this function the Rolle's theorem will be applicable
a.Any inetrval
b.$\small [0,\frac{\pi}{2}]$
c.$\small [0,\pi]$
d.$\small (0,\pi)$
Ans:d
Q7.$\small \phi(x)=1-x^{\frac{2}{3}}$ for this function the Rolle's theorem will be applicable
a.Any inetrval
b.$\small [-1,1]$
c.$\small [-1,0]$
d.$\small (0,1)$
Ans:b
Q8.$\small y=|x|$ this function satisfy how many conditions of Rolle's Theorem in this interval $\small [-1,1]$
a.one condition
b.two condition
c.all condition
d.No condition
Ans:a,d
Q9.Find the value of $\small c$,If $\small f(x)=x^3-3x-1$ function satisfy all the condition of Lagrange's M-V theorem in this interval $\small [\frac{-11}{7},\frac{13}{7}]$
a.2
b.-1
c.0
d.1
Ans:b,d
Q10.Rolle's theorem will be applicable for $\small f(x)=x^3+bx^2+ax+5$ in this interval [1,3] when $\small c=2+\frac{1}{\sqrt 3}$
a.a=-11
b.a=11
c.b=6
d.b=-6
Ans:b,d
Q11.Rolle's theorem will be applicable for $\small f(x)=x^3+px^2-qx+4$ in this interval [-2,2] when $\small c=\frac{1}{3}(1+\frac{1}{\sqrt 13})$
a.p=-1
b.p=1
c.q=4
d.q=3
Ans:a,c
Q12.If$\small f(x)=(x-1)(x-2)(x-3)(x-4)$,$\small f^{'}(x)=0$ has three equation $\small \alpha,\beta,\gamma$
a.$\small 1<\alpha<2$
b.$\small 2<\beta<3$
c..$\small 3<\gamma<4$
d..$\small 4<\alpha<5$
Ans:a,b,c
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