Exercise-2

Let $x_1<\alpha <\beta <\gamma <x_4, x_1<x_2<x_3$. If $f\left(x\right)$ is a cubic polynomial with real coefficients such that $\left(f\right(\alpha ){)}^2+(f\left(\beta \right){)}^2+\left(f\right(\gamma ){)}^2=0, f\left(x_1\right)f\left(x_2\right)<0, f\left(x_2\right)f\left(x_3\right)<0$ and $f\left(x_1\right)f\left(x_3\right)>0$, then which of the following are CORRECT?

If $f\left(x\right)$ is cubic polynomial with real coefficients, $\alpha <\beta <\gamma$ and $x_1<x_2$ be such that $f\left(\alpha \right)=f\left(\beta \right)=f\left(\gamma \right)=f\text{'}\left(x_1\right)=f^{\text{'}}\left(x_2\right)=0$, then possible graph of $y=f\left(x\right)$ is: (assuming $y$-axis vertical)

If a differentiable function $f$ satisfies $f\left(\frac{x+y}{3}\right)=\frac{4-2\left(f\left(x\right)+f\left(y\right)\right)}{3} \forall x, y\in R$, then $f\left(x\right)$ is equal to

If $f\left(x\right)=a_0+\sum _{k=1}^n{a}_k{\left|x\right|}^k$, where $a_i^{\text{'}}s$ are real constants, then $f\left(x\right)$ is

Let $f:R\to R$ be a function such that $f\left(0\right)=1$ and for any $x,y\in R, f\left(xy+1\right)=f\left(x\right)f\left(y\right)-f\left(y\right)-x+2$. Then, $f$ is
 

Suppose that $f$ is a differentiable function with the property that $f\left(x+y\right)=f\left(x\right)+f\left(y\right)+xy$ and $\underset{h\to 0}{\lim }\frac{1}{h}f\left(h\right)=3$ (where $\left[.\right]$ represents greatest integer function), then

Let $f$ be a real valued function defined for all real numbers $x$ such that for some positive constant $a$, the equation $f\left(x+a\right)=\frac{1}{2}+\sqrt{f\left(x\right)-{\left(f\left(x\right)\right)}^2}$ holds for all $x$. Then $f\left(x\right)$ is periodic function with period equal to

If $f\left(x\right)=(x-4)(x-5)(x-6)(x-7)$, then,