Question

Consider the following statements
$S_1:$ If $x^3+y^2=5(xy-1),$ then $\frac{dy}{dx}$ at $(1,3)$ is equal to $12$
$S_2:$ If $x^e=e^y,$ then $\frac{dy}{dx}$ at $x=1$ is equal to $e$
$S_3:$ If $f\left(x\right)$ is a linear polynomial, then $f^{\prime }\left(\sin x\right)+f^{\prime \prime }\left(\sin x\right)$ is constant.
Which of the following is/are correct about the truth value of above statements

• A. $S_1$ is false and $S_3$ is true
• B. $S_2$ and $S_3$ both are true
• C. $S_1$ and $S_3$ both are true
• D. $S_1$ is true and $S_2$ is false

Answer 1

Answer: (B), (C)

$S_1$ and $S_3$ both are true

$S_1:3x^2+2y{y}^{\prime }=5(y+x{y}^{\prime })$
at $(1,3);3+6y^{\prime }=5(3+y^{\prime })$
$\Rightarrow y^{\prime }=12$

$S_2:x^e=e^y$
$e\ln x=y$
$\Rightarrow y^{\prime }=\frac{e}{x}$
at $x=1,y^{\prime }=e$

$S_3:f\left(x\right)=ax+b\Rightarrow f\left(\sin x\right)=a\sin x+b$
$f^{\prime }\left(\sin x\right)\cdot \cos x=a\cos x\Rightarrow f^{\prime }\left(\sin x\right)=a$
$f^{\prime \prime }\left(\sin x\right)=0$
$\Rightarrow f^{\prime \prime }\left(\sin x\right)+f^{\prime }\left(\sin x\right)=a$ (constant)