[amp_mcq option1=”Local minimum” option2=”Neither local minimum nor a local maximum” option3=”Local maximum” option4=”Both local minimum and local maximum” correct=”option1″]
The correct answer is A. Local minimum.
A local minimum is a point in a function’s domain where the function’s value is less than or equal to the values of the function in all neighboring points. In other words, a local minimum is a point where the function “bottoms out.”
The function $f(x, y) = x^2 – y^2$ is defined on the entire plane $R^2$. To find the local minima of $f$, we need to find the points where its gradient is equal to zero. The gradient of $f$ is given by the vector $(2x, -2y)$.
The gradient of $f$ is equal to zero at the point $(0, 0)$. Therefore, $(0, 0)$ is a critical point of $f$. To determine whether $(0, 0)$ is a local minimum or a local maximum, we need to evaluate $f$ at points in the neighborhood of $(0, 0)$.
The value of $f$ at the point $(1, 0)$ is $1$. The value of $f$ at the point $(0, 1)$ is $-1$. Therefore, we can see that the value of $f$ decreases as we move away from $(0, 0)$ in the direction of $(1, 0)$, and the value of $f$ increases as we move away from $(0, 0)$ in the direction of $(0, 1)$. This tells us that $(0, 0)$ is a local minimum.
In conclusion, the point $(0, 0)$ is a local minimum of the function $f(x, y) = x^2 – y^2$.