Question

The function of two variables is maximum if:
A. \(rt-s^2>0\),
B. \(r<0\),
C. \(rt-s^2<0\),
D. \(r>0\), E. \(rt-s^2=0,\ r<0\).

• A. A, B
• B. B, C
• C. C, D
• D. A, B, E

Hint:

For maximum of \(f(x,y)\), use \(rt-s^2>0\) and \(r<0\).

Answer 1

The Correct Option is A

Concept:
For a function of two variables, second derivative test is used to identify maximum and minimum. Let: \[ r=f_{xx},\quad s=f_{xy},\quad t=f_{yy} \] The discriminant is: \[ D=rt-s^2 \]

Step 1: Condition for maximum.
For maximum: \[ D=rt-s^2>0 \] and: \[ r=f_{xx}<0 \]

Step 2: Match with statements.
Statement A: \[ rt-s^2>0 \] is correct. Statement B: \[ r<0 \] is also correct.

Step 3: Final answer.
Thus, the function has maximum when: \[ rt-s^2>0,\quad r<0 \] \[ \therefore \text{Correct Answer is (A)} \]