Finding Derivative of a Function

Q. Value of $\underset{x\to 2}{lim}(8-3x+12x^2)$ is

Q. Let $f\left(x\right)$ be a polynomial of degree $6,$ which satisfies $\underset{x\to 0}{lim}{(1+\frac{f\left(x\right)}{x^3})}^{\frac{1}{x}}=e^2$ has local maximum at $x=1$ and local minimum at $x=0$ and $2,$ then $\frac{5f\left(3\right)}{4}$ is

Q. If $f\left(x\right)$ is a quadratic polynomial with leading coefficient $1$ such that $\underset{x\to 0}{lim}f\left(x\right)=3=\underset{x\to 0}{lim}{f}^{\prime }\left(x\right)$, then $\underset{x\to 2}{lim}f\left(x\right)=$
$\left(f^{\prime }\right(x)=\frac{df\left(x\right)}{dx})$

1. $\underset{x\to 1}{lim}f\left(2x\right)$
2. $\underset{x\to -5}{lim}f\left(x\right)$
3. $\underset{2x\to -5}{lim}f\left(2x\right)$
4. $13$

Q. If $\underset{x\to -1}{lim}(a{x}^2+bx+c)=1$ where $a,b,c\in R,$ then which of the following is true

1. $a-b+c=1$
2. $a+b+c=1$
3. $a+b-c=1$
4. $a+b+c+1=0$

Q. Let $P\left(x\right)$ be a polynomial satisfying $\underset{x\to \infty }{lim}\frac{x^{3}P\left(x\right)}{x^6+3x^2+7}=2.$ If $P\left(1\right)=2,$ $P\left(3\right)=10$ and $P\left(5\right)=26,$ then the value of $\frac{P\left(2\right)+|P\left(0\right)|}{10}$ is

Q.

1. $14$
2. $31$
3. it does not exist
4. $18$

Q. If $f\left(x\right)$ is a polynomial of degree $4$ such that $\underset{x\to -1}{lim}\frac{f\left(x\right)}{(x+1{)}^3}=1$ and $f^{\prime \prime \prime }\left(0\right)=-12$, then the maximum value of $f\left(x\right)$ is
​​​​​​​(correct answer + 1, wrong answer - 0.25)

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

Q. If $f\left(x\right)$ is a polynomial and $\underset{t\to a}{lim}\frac{\underset{a}{\overset{t}{\int }}f\left(x\right)dx-\frac{(t-a)}{2}\left(f\right(t)+f(a\left)\right)}{(t-a{)}^3}=0$ for all $a$, then the degree of $f\left(x\right)$ can atmost be

Q. If $\underset{x\to 3}{lim}(3k+2x^2)=0,$ then $k$ is equal to

1. $-6$
2. $6$
3. $1$
4. $-1$

Q. If $f\left(x\right)$ is a polynomial of degree $4$ such that $\underset{x\to -1}{lim}\frac{f\left(x\right)}{(x+1{)}^3}=1$ and $f^{\prime \prime \prime }\left(0\right)=-12$, then the maximum value of $f\left(x\right)$ is
​​​​​​​(correct answer + 1, wrong answer - 0.25)

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

Q.

$\underset{x\to \infty }{lim}\frac{(2x+1{)}^{40}(4x-1{)}^5}{(2x+3{)}^{45}}$

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Q. There are $x^4+57x+15$ pens to be distributed in a class of $x^2+4x+2$ students. Each student should get the minimum possible number of pens. Find the number of pens received by each student and the number of pens left undistributed $\left(xϵN\right)$.

1. $9x-13$
2. $9x-15$
3. $9x-20$
4. $9x+13$

Q. If $f\left(x\right)$ is a polynomial of degree $4$ such that $\underset{x\to -1}{lim}\frac{f\left(x\right)}{(x+1{)}^3}=1$ and $f^{\prime \prime \prime }\left(0\right)=-12$, then the maximum value of $f\left(x\right)$ is
​​​​​​​(correct answer + 1, wrong answer - 0.25)

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

Q. Let $f\left(x\right)=\{\begin{array}{cc}{x}^2-1,& 0<x<2\\ 2x+3,& 2\le x<3\end{array}$
If $\alpha =\underset{x\to 2^{-}}{lim}f\left(x\right)$ and $\beta =\underset{x\to 2^{+}}{lim}f\left(x\right),$ then the quadratic equation whose roots are $\alpha ,\beta$ is

1. $x^2+10x+21=0$
2. $x^2-10x+21=0$
3. $x^2-7x+12=0$
4. $x^2-12x+10=0$