Decide if the statements are true, or false. (a) If f(x, y) has a local minimum at P_0, then the...

Question:

Decide if the statements are true, or false.

(a) If {eq}f(x, y) {/eq} has a local minimum at {eq}P_0 {/eq}, then the function {eq}g(x, y) = -f(x, y) {/eq} has a local maximum at {eq}P_0 {/eq}.

(b) If {eq}P_0 {/eq} is a local maximum of {eq}f {/eq}, then {eq}f(a, b) < f(P_0) {/eq} for all points {eq}(a, b) {/eq} in 2-space.

(c) If {eq}P_0 {/eq} is a critical point of {eq}f {/eq}, then {eq}P_0 {/eq} is either a local maximum or local minimum.

Extrema of a Function:

{eq}\\ {/eq}

Local Mamima/Minima - Local Maxima and Local Minima are defined only in a particular range of a function, they do not cover the entire domain. Local minima is the smallest value of the function in a given range whereas Local maxima is the largest value of the function in the given range.

Saddle point - Saddle point of a function is such a point which is neither a local maxima nor a local minima. It is also known as a Minimax point.

Answer and Explanation:

{eq}\\ {/eq}

(a) TRUE

Explanation - Since, local minimum is point where the double derivative is greater than zero in the given range of that...

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