The critical point and nature for the function is

Minimum Explanation: Concept: A critical point of a function of is a value in the domain of where it is not differentiable or its derivative is . To find the nature of the critical points, we apply the second derivative test. I. If is less than , then the given function is said to be maxima. II. If is greater than , then the function is said to be minima. Calculation: Here, Partial derivatives, and Now, for critical points, ⇒ ⇒ , Also, ⇒ ⇒ , So, critical points and So, at minimum Hence, option (4) is correct.

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NIMCET 2017 — Mathematics PYQ

NIMCET | Mathematics | 2017

The critical point and nature for the function $f (x, y) = x^{2} - 2 x + 2 y^{2} + 4 y - 2$ is

Choose the correct answer:

Explanation

Concept:
A critical point of a function of $f (x)$ is a value $x_{0}$ in the domain of $f$ where it is not differentiable or its derivative is $0$ $(f^{'} (x_{0}) = 0)$ .

To find the nature of the critical points, we apply the second derivative test.

I. If $f^{''} (x)$ is less than $0$ , then the given function is said to be maxima.
II. If $f^{''} (x)$ is greater than $0$ , then the function is said to be minima.

Calculation:
Here, $f(x, y) = x² - 2x + 2y² + 4y - 2$
Partial derivatives,
$f^{'} (x) = 2 x - 2$ and $f^{'} (y) = 4 y + 4$
Now, for critical points, $f^{'} (x) = 0$
⇒ $2 x - 2 = 0$
⇒ $x = 1$ ,
Also, $f^{'} (y) = 0$
⇒ $4 y + 4 = 0$
⇒ $y = - 1$ ,
So, critical points $(1, - 1)$
$f''(x) = 2 > 0$ and $f''(y) = 4 > 0$
So, at $(1, - 1)$ minimum
Hence, option (4) is correct.