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Curve with Concavity Proportional to log x | JEE DE

JEE Maths question with a full step-by-step solution.

Question

A curve, whose concavity is directly proportional to the logarithm of its

x

-coordinate at any point of the curve, is given by:

C_1 x^2(2\log x - 3) + C_2 x + C_3

correct

C_1 x^2(2\log x + 3) + C_2 x + C_3

C_1 x^2(2\log x) + C_2

DNone of these

Solution

Concavity means the second derivative, so the condition is

\frac{d^2y}{dx^2} = K\ln x.

Step 1: Integrate once (by parts):

\frac{dy}{dx} = K\int\ln x\,dx = K(x\ln x - x) + c.

Step 2: Integrate again (by parts on

\int x\ln x\,dx = \frac{x^2}{2}\ln x - \frac{x^2}{4}

y = K\left(\frac{x^2}{2}\ln x - \frac{x^2}{4} - \frac{x^2}{2}\right) + cx + d = K\left(\frac{x^2}{2}\ln x - \frac{3x^2}{4}\right) + cx + d.

Step 3: Factor:

y = \frac{K}{4}x^2(2\ln x - 3) + cx + d = C_1 x^2(2\log x - 3) + C_2 x + C_3.

Correct answer: (1)

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