Proof of Intermediate Value Theorem

Q.

Factories:
$x^3+13x^2+32x+20$

Q.

If $x^2$ - x + a - 3 < 0 for at least one negative value of x, then complete set of values of a

1. (

2. (

3. (

4. (

Q. Let $f:R\to R$ is differentiable and strictly increasing function throughout its domain.
$S_1:$ If $|f\left(x\right)|$ is also strictly increasing function, then $f\left(x\right)=0$ has no real roots.
$S_2:$ At $\infty$ or $-\infty ,f\left(x\right)$ may approach to $0,$ but it can't be equal to zero

1. $S_1$ is true, $S_2$ is true, $S_2$ is correct explanation of $S_1$
2. $S_1$ is true, $S_2$ is true, $S_2$ is not correct explanation of $S_1$
3. $S_1$ is true, $S_2$ is false.
4. $S_1$ is false, $S_2$ is true.

Q. Let $f\left(x\right)$ be a function such that $f\left(a_1\right)=0,f\left(a_2\right)=1,$ $f\left(a_3\right)=-2,f\left(a_4\right)=3$ and $f\left(a_5\right)=0;$ where $a_i\in R$ and $a_i<a_j\forall i<j.$ Let $g\left(x\right)$ be a function defined as $g\left(x\right)=f^{\prime }(x{)}^2+f(x\left)f^{\prime \prime }\right(x)$ on $[a_1,a_5].$ If $f\left(x\right)$ be thrice differentiable, then $g\left(x\right)=0$ has at least

1. $6$ real roots
2. $4$ real roots
3. $7$ real roots
4. $5$ real roots

Q. If the roots of $x^2+bx+c=0$ are two consecutive integers, then $b^2-4c=$

1. $1$
2. $3$
3. $0$
4. $2$

Q.

If a $ϵ$ R- and a $\ne$ - 2 then the equaiton $x^2+a|x|+1=0:$

1. Must have exactly two real roots

2. Can not have any real root

3. Must have either four real roots or no real roots

4. Must have either exactly two real roots or no real roots

Q. Let $g\left(x\right)$ is a continuous and differentiable function and $f\left(x\right)=x\cdot g\left(x\right)$ has a positive root at $x=\alpha ,$ then one root of $f^{\prime }\left(x\right)=0$ can be

1. $-\frac{\alpha }{2}$
2. $\frac{\alpha }{2}$
3. $\frac{\alpha }{3}$
4. $\frac{3\alpha }{2}$

Q. The function $f\left(x\right)=\frac{4-x^2}{4x-x^3}$ is

1. discontinuous at only one point
2. discontinuous exactly at two points
3. discontinuous exactly at three points
4. None of these

Q. $x^2-10ax-11b=0roots=c,d$
$x^2-10cx-11d=0roots=a,b$ find a + b + c + d.

Q.

The condition for (a - 2) $x^2$ + 2ax + (a + 3) = 0 to have both roots to be real is

1. a [2, )

2. a (- , -3]

3. a (-, 6]

4. a (-2, 3]

Q. Let $f\left(x\right)=a_0+a_1{x}^2+a_2{x}^4+\dots .+a_n{x}^{2n}$ when $0<a_0<a_1<\dots \dots <a_n$ then $f\left(x\right)$ has

1. No extremum
2. Only one maximum
3. Only one minimum
4. Two maximums

Q. Let $f\left(x\right)$ be a function such that $f\left(a_1\right)=0,f\left(a_2\right)=1,$ $f\left(a_3\right)=-2,f\left(a_4\right)=3$ and $f\left(a_5\right)=0;$ where $a_i\in R$ and $a_i<a_j\forall i<j.$ Let $g\left(x\right)$ be a function defined as $g\left(x\right)=f^{\prime }(x{)}^2+f(x\left)f^{\prime \prime }\right(x)$ on $[a_1,a_5].$ If $f\left(x\right)$ be thrice differentiable, then $g\left(x\right)=0$ has at least

1. $4$ real roots
2. $5$ real roots
3. $6$ real roots
4. $7$ real roots

Q. Let $f\left(x\right)$ be a function such that $f\left(a_1\right)=0,f\left(a_2\right)=1,$ $f\left(a_3\right)=-2,f\left(a_4\right)=3$ and $f\left(a_5\right)=0;$ where $a_i\in R$ and $a_i<a_j\forall i<j.$ Let $g\left(x\right)$ be a function defined as $g\left(x\right)=f^{\prime }(x{)}^2+f(x\left)f^{\prime \prime }\right(x)$ on $[a_1,a_5].$ If $f\left(x\right)$ be thrice differentiable, then $g\left(x\right)=0$ has at least

1. $4$ real roots
2. $5$ real roots
3. $6$ real roots
4. $7$ real roots

Q.
Let $f\left(x\right)$ and $g\left(x\right)$ are differentiable function such that $g\left(x\right)=2xf\left(x\right)+x^2{f}^{\prime }\left(x\right)$ in $[a,d]$ and $0<a<b<c<d,f\left(a\right)=0,f\left(b\right)=5,f\left(c\right)=-3,f\left(d\right)=0,$ then the minimum number of zero(s) for $g\left(x\right)=0$ is