Derivative of Standard Functions

Q. If for $f\left(x\right)={\mathrm{ax}}^2+\mathrm{bx}+12,f\text{'}\left(4\right)=15\mathrm{and}f\text{'}\left(2\right)=11,$ then a + b =

Q. If $f\left(x\right)=3x^2+5x-7$, then the value of $f\text{'}\left(0\right)+3f\text{'}(-1)$ is

Q. Let $f:R\to R$ be defined by $f\left(x\right)=\frac{x^2-3x-6}{x^2+2x+4}$. Then which of the following statements is(are) TRUE?

1. $f$ is decreasing in the interval $(-2,-1)$
2. Range of $f$ is $[-\frac{3}{2},2]$
3. $f$ is onto
4. $f$ is increasing in the interval $(1,2)$

Q.

$If|x|<1then\frac{d}{dx}(1-2x+3x^2-........)=$

1. $\frac{-2}{(1+x{)}^3}$

2. $\frac{2}{(1-x{)}^2}$

3. $\frac{-3}{(1+x{)}^4}$

4. $\frac{2}{(1+x{)}^3}$

Q.

$\frac{d}{dx}\left\{x^{1/x}\right\}=$

1. $x^{1/x-2}(1+logx)$

2. $x^{\frac{1}{x}-2}(1-logx)$

3. 1+logx

4. $1+log{x}^{1/x-2}$

Q.

If $y^2$ = $a{x}^2+bx+c$, then $y^3.\frac{d^{2}y}{d{x}^2}$ is

1. a constant

2. a function of x only

3. a function of y only

4. a function of x and y

Q.

If $y$ = $(1+x)(1+x^2)(1+x^4)....(1+x^{2n})$, then $\frac{dy}{dx}$ at x = 0 is

1. 0

2. -1

3. 1

4. 2

Q. In $△ABC,$ sides opposite to angles $A,B,C$ are denoted by $a,b,c$ respectively. Then the value of $(b-c)\sin A+(c-a)\sin B+(a-b)\sin C=$

1. $a^2+b^2+c^2$
2. $a+b+c$
3. $ab+bc+ca$
4. $0$

Q.

If $y^2$ = $a{x}^2+bx+c$, then $y^3.\frac{d^{2}y}{d{x}^2}$ is

1. a constant

2. a function of x only

3. a function of y only

4. a function of x and y

Q.

$\frac{d}{dx}\left\{x^{1/x}\right\}=$

1. $x^{1/x-2}(1+logx)$

2. $x^{\frac{1}{x}-2}(1-logx)$

3. 1+logx

4. $1+log{x}^{1/x-2}$

Q. If $f\left(x\right)=x\sin \left(\frac{1}{x}\right),x\ne 0$ is continuous at $x=0,$ then the value of $f\left(0\right)$ is

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