GATE CSE 1993 | Question: 01.7

Question

The function $f\left(x,y\right) = x^2y - 3xy + 2y +x$ has

• A. no local extremum
• B. one local minimum but no local maximum
• C. one local maximum but no local minimum
• D. one local minimum and one local maximum

Answer 1

Answer: A

$r = \frac{\partial^2 f}{\partial x^2} = 2y$

$s = \frac{\partial^2 f}{\partial x \partial y} = 2x - 3$

$t = \frac{\partial^2 f}{\partial y^2} = 0$

Since, $rt - s^2 \leq 0,$ (if < 0 then we have no maxima or minima, if = 0 then we can't say anything).

Maxima will exist when $rt - s^2 > 0$ and $r < 0.$

Minima will exist when $rt - s2 > 0$ and $r > 0.$

Since, $rt - s^2$ is never $> 0$ so we have no local extremum.

Comments

When x = 1.5, s² = 0 and 0 - 0 = 0. 

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Corrected.

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multivariable calculus not in syllabus?