Match List-I with List-II.
| List-I | List-II | ||
| (A) | If $y=log_e\left(\frac{x^2}{e^3}\right),$ then $\frac{d^2y}{dx^2}$ is, | (I) | $x^2(7+12log_ex)$ |
| (B) | If $y=x^4log_ex,$ then $\frac{d^2y}{dx^2}$ is, | (II) | $\frac{2log_ex-3}{x^3}$ |
| (C) | If $y=x^3e^x,$ then $\frac{d^2y}{dx^2}$ is, | (III) | $\frac{-2}{x^2}$ |
| (D) | If $y=\frac{log_ex}{x},$ then $\frac{d^2y}{dx^2}$ is, | (IV) | $xe^x(x^2+6x+6)$ |
Choose the correct answer from the options given below :
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A)-(III),(B)-(I),(C)-(IV),(D)-(II)
(A) $y=\log_e\left(\frac{x^2}{e^3}\right)⇒\frac{dy}{dx}=\frac{e^3}{x^2}×\frac{2x}{e^3}=\frac{2}{x}⇒\frac{d^2y}{dx^2}=\frac{-2}{x^2}$ (III)
(B) $y=x^4\log_ex⇒\frac{dy}{dx}4x^3\log_ex+x^3⇒\frac{d^2y}{dx^2}=12x^2\log x+4x^2+3x^2=12x^2\log x+7x^2$ (I)
(C) $y=x^3e^x⇒\frac{dy}{dx}=3x^2e^x+x^3e^x⇒\frac{d^2y}{dx^2}=6xe^x+3x^2e^x+x^3e^x+3x^2e^x$ (IV)
(D) $y=\frac{\log_ex}{x}⇒\frac{dy}{dx}=\frac{\frac{1}{x}×x+\frac{\log x}{x^2}}{x^2}⇒\frac{d^2y}{dx^2}=\frac{2\log x-3}{x^3}$ (II)