Geometrical Explanation of Intermediate Value Theorem

Q. Set of values for $p$ for which the function given by $f\left(x\right)=x^3-2x^2-px+1$ is one-one function $\forall x\in R$ is

1. $(-\infty ,\frac{-4}{3})$
2. $(\frac{-4}{3},\infty )$
3. $(\frac{-4}{3},\frac{4}{3})$
4. $R$

Q. Let $y=f\left(x\right)$ be a polynomial function whose degree is greater than zero such that $f\left(\alpha \right)$ and $f\left(\frac{1}{\alpha }\right)$ satisfy the equation $x^3-(1-a)x^2-2ax+a=0$ $\forall \alpha \in R-\left\{0\right\}$ where $a\in R$. If ${\begin{array}{c}\frac{d^{4}y}{d{x}^4}\end{array}|}_{x=2}=0$ and ${\begin{array}{c}\frac{d^{3}y}{d{x}^3}\end{array}|}_{x=2}=-6,$ then for $\alpha =2,$ the value of $8a$ is

Q. Let $F\left(x\right)=1+f\left(x\right)+{\left(f\right(x\left)\right)}^2+{\left(f\right(x\left)\right)}^3,$ where $f\left(x\right)$ is an increasing differentiable function and $F\left(x\right)=0$ has a positive root, then

1. $F\left(x\right)$ is an increasing function
2. $F\left(0\right)\le 0$
3. $f\left(0\right)\le -1$
4. $F^{\prime }\left(0\right)\ge 0$

Q. If the function $f\left(x\right)=a{x}^3+b{x}^2+11x-6,a,b\in R$ satisfies Rolle's theorem in $[1,3]$ and $f^{\prime }(2+\frac{1}{\sqrt{3}})=0,$ then the value of $(a-b)$ is

Q. Set of values for $p$ for which the function given by $f\left(x\right)=x^3-2x^2-px+1$ is one-one function $\forall x\in R$ is

1. $R$
2. $(\frac{-4}{3},\infty )$
3. $(\frac{-4}{3},\frac{4}{3})$
4. $(-\infty ,\frac{-4}{3})$

Q. If $27a+9b+3c+d=0$, then the equation $4a{x}^3+3b{x}^2+2cx+d=0$ has atleast one real root lying in

1. $(0,1)$
2. $(1,3)$
3. $(0,3)$
4. $(2,4)$

Q. Which of the following statements is/are true for the function $f\left(x\right)=x^{13}+7x^3-5x+1$

1. $f\left(x\right)$ has at least $1$ root (real) in $[0,1]$
2. $f^{\prime }\left(x\right)$ has at least $1$ root (real) in $R$
3. $f\left(x\right)$ has at most $1$ root (real) in $[0,1]$
4. $f^{\prime }\left(x\right)$ has at most $1$ real root in $R$

Q. Which of the following statements is/are true for the function $f\left(x\right)=x^{13}+7x^3-5x+1$

1. $f\left(x\right)$ has at least $1$ root (real) in $[0,1]$
2. $f^{\prime }\left(x\right)$ has at least $1$ root (real) in $R$
3. $f\left(x\right)$ has at most $1$ root (real) in $[0,1]$
4. $f^{\prime }\left(x\right)$ has at most $1$ real root in $R$

Q. Which of the following statements is/are true for the function $f\left(x\right)=x^{13}+7x^3-5x+1$

1. $f\left(x\right)$ has at least $1$ root (real) in $[0,1]$
2. $f^{\prime }\left(x\right)$ has at least $1$ root (real) in $R$
3. $f\left(x\right)$ has at most $1$ root (real) in $[0,1]$
4. $f^{\prime }\left(x\right)$ has at most $1$ real root in $R$