Intermediate Value Theorem

Q. The number of solutions(s) of the equation $3\tan x+x^3=2\forall x\in \left(0,\frac{\pi }{4}\right)$ is .

Q. Verify Rolle's theorem for the function $y=x^2+2,a=-2$ and $b=2$.

Q.

Find the number of solutions of cos(x)+cos(2x)+cos(3x)+cos(4x)+cos(5x)=5 in the interval [0, 2π]

Q.

Intermediate value theorem states that if a function, f, with an interval, [a, b], as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval

Q.

There is a solution to $x^5-2x^3-2$ = 0 between x=0 and x=2

1. False

2. True

Q. Verify Rolle's theorem for the function $f\left(x\right)=x^2+2x-8,x\in [-4,2]$

Q.

The number of real solutions of the equation

cos(e^x) = 2^x + 2^-x is

(a) 0 (b) 1 (c) 2 (d) infinitely many

Q. The values of $a,m$ and $b$, for which Lagranges mean value theorem is applicable to the function $f\left(x\right)$ for $x\in [0,2]$ is

$f\left(x\right)=\{\begin{array}{cc}3,& x=0\\ -x^2+a,& 0<x<1\\ mx+b,& 1\le x\le 2\end{array}$

1. $a=3,m=-2,b=0$
2. No such, $a,m,b$ exist
3. $a=3,m=2,b=0$
4. $a=3,m=-2,b=4$

Q. The number of real solution of the equation \operatorname{tan^{-1\sqrt{x^2-3x+7+\operatorname{cos^{-1\sqrt{4x^2-x+3=π is

Q.

Intermediate value theorem states that if a function, f, with an interval, [a, b], as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval

1. True

2. False

Q. the number of real solutions of equation is:
sqrt(1 + cos2x) = sqr(2).sin^-1(sinx)
x belongs to [-pi, pi]
options: 0, 1, 2, infinite

Q. Differentiation measures how fast the function is changing with respect to the change in independent variable.

1. True
2. False

Q. For all twice differentiable functions $f:R\to R$, with $f\left(0\right)=f\left(1\right)=f^{\prime }\left(0\right)=0$,

1. $f^{\prime \prime }\left(x\right)\ne 0$, at every point $x\in (0,1)$
2. $f^{\prime \prime }\left(0\right)=0$
3. $f^{\prime \prime }\left(x\right)=0$, for some $x\in (0,1)$
4. $f^{\prime \prime }\left(x\right)=0$, at every point $x\in (0,1)$

Q.

There is a solution to $x^5-2x^3-2=0$ between x=0 and x=2

1. T

Q.

There is a solution to $x^5-2x^3-2=0$ between x=0 and x=2

1. T

Q. The number of solutions(s) of the equation $3\tan x+x^3=2\forall x\in \left(0,\frac{\pi }{4}\right)$ is .

Q. Let X₁ and X₂ are optimal solutions of a LPP, then
(a) X = λ X₁ + (1 − λ) X₂, λ ∈ R is also an optimal solution
(b) X = λ X₁ + (1 − λ) X₂, 0 ≤ λ ≤ 1 gives an optimal solution
(c) X = λ X₁ + (1 + λ) X₂, 0 ≤ λ ≤ 1 gives an optimal solution
(d) X = λ X₁ + (1 + λ) X₂, λ ∈ R gives an optimal solution