Applications of Derivatives Question - None

Question

Paragraph for Question nos. 583 to 584
Let $y = f(x)$ be a differentiable function passing through $(1, 0)$. Let slope of the tangent at the point $(x, f(x))$ be $m_1$ and slope of line joining the point and origin be $m_2$. Also $\left|\dfrac{\log(m_1 + m_2)}{\log x}\right| = \dfrac{2}{1}$.

If $f_1(x)$ and $f_2(x)$ are 2 functions satisfying the above property where $f_1(x)$ is an algebraic function and $f_2(x)$ is a transcendental function.

Which one of the following statement is correct?

Accepted Answer

Step 1: Identify the slopes. Given m₁ = f'(x) (slope of tangent) and m₂ = f(x)/x (slope from origin to point (x, f(x))).

Step 2: Solve the given condition. From |log(m₁ + m₂)/log x| = 2, we have two cases:
Case 1: log(m₁ + m₂) = 2log x ⟹ m₁ + m₂ = x²
Case 2: log(m₁ + m₂) = -2log x ⟹ m₁ + m₂ = x⁻²

Step 3: Form differential equations.
Case 1: f'(x) + f(x)/x = x² ⟹ This is linear DE
Case 2: f'(x) + f(x)/x = 1/x² ⟹ This is also linear DE

Step 4: Solve using integrating factor μ(x) = x.
Case 1: d/dx[xf(x)] = x³ ⟹ xf(x) = x⁴/4 + C. Using f(1) = 0: C = -1/4, so f₁(x) = x³/4 - 1/(4x) (algebraic)
Case 2: d/dx[xf(x)] = 1/x ⟹ xf(x) = ln x + C. Using f(1) = 0: C = 0, so f₂(x) = ln x/x (transcendental)

Step 5: Both functions satisfy the given differential conditions and initial condition. f₁(x) is algebraic and f₂(x) is transcendental as required.

∴ Answer: D

Question:

Step-by-Step Solution

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