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Question

Differential Equations Question on Differential Equations

Let f(x, y) = ln(1 + x2 + y2 ) for (x, y) ∈ R2\R^2. Define
P=∂2f∂x2∣(0,0)Q=∂2f∂x∂y∣(0,0) R=∂2f∂y∂x∣(0,0)S=∂2f∂y2∣(0,0)\begin{matrix} P=\frac{∂^2f}{∂x^2}|_{(0,0)} & Q=\frac{∂^2f}{∂x∂y}|_{(0,0)} \\\ R=\frac{∂^2f}{∂y∂x}|_{(0,0)} & S=\frac{∂^2f}{∂y^2}|_{(0,0)} \end{matrix}
Then

A

PS − QR > 0 and P < 0

B

PS − QR > 0 and P > 0

C

PS − QR < 0 and P > 0

D

PS − QR < 0 and P < 0

Answer

PS − QR > 0 and P > 0

Explanation

Solution

The correct option is (B) : PS − QR > 0 and P > 0.