Question

If \( Z = f(x + ay) + \phi(x - ay) \), then

• A. \( Z_{xx} = Z_{yy} \)
• B. \( Z_{xx} = a^2 Z_{yy} \)
• C. \( Z_{yy} = a^2 Z_{xx} \)
• D. None of these

Hint:

When differentiating functions of the form \( f(x \pm ay) \), use the chain rule and remember to apply the appropriate factor when differentiating with respect to \( x \) or \( y \).

Answer 1

The Correct Option is C

Step 1: Express the function \( Z \).
We are given the function: \[ Z = f(x + ay) + \phi(x - ay) \] To find the second partial derivatives of \( Z \), we will differentiate with respect to \( x \) and \( y \).

Step 2: Differentiate \( Z \) with respect to \( x \).
First, compute the partial derivative of \( Z \) with respect to \( x \): \[ \frac{\partial Z}{\partial x} = f'(x + ay) + \phi'(x - ay) \] Now, take the second derivative with respect to \( x \): \[ Z_{xx} = f''(x + ay) + \phi''(x - ay) \]

Step 3: Differentiate \( Z \) with respect to \( y \).
Next, compute the partial derivative of \( Z \) with respect to \( y \): \[ \frac{\partial Z}{\partial y} = a f'(x + ay) - a \phi'(x - ay) \] Now, take the second derivative with respect to \( y \): \[ Z_{yy} = a^2 f''(x + ay) + a^2 \phi''(x - ay) \]

Step 4: Compare \( Z_{xx} \) and \( Z_{yy} \).
From the expressions for \( Z_{xx} \) and \( Z_{yy} \), we observe: \[ Z_{yy} = a^2 Z_{xx} \] Thus, the correct answer is \( Z_{yy} = a^2 Z_{xx} \), which corresponds to option (C).