Question

If \( y = f(x) \) satisfies the differential equation \[ (x^2 - 4)y' - 2xy + 2x(4 - x^2)^2 = 0 \] and \( f(3) = 15 \), then find the local maximum value of \( f(x) \):

• A. 16
• B. 20
• C. 25
• D. 30

Hint:

Local extrema of a function occur where its first derivative is zero or undefined.

Answer 1

The Correct Option is A

Step 1: Rearrange the differential equation.
Given \[ (x^2 - 4)y' - 2xy + 2x(4 - x^2)^2 = 0 \] \[ \Rightarrow (x^2 - 4)y' = 2x\big[y - (4 - x^2)^2\big] \]
Step 2: Identify critical points.
At extrema, \( y' = 0 \). Hence, \[ y = (4 - x^2)^2 \]
Step 3: Find the extremum point.
Substitute \( y = (4 - x^2)^2 \) into the condition \( f(3) = 15 \) to determine constants and the nature of extremum.
Step 4: Evaluate the local maximum value.
At the critical point, the function attains its local maximum value: \[ f(x) = 16 \]