Question

Determine the signs of the partial derivatives for the function f whose graph is shown

(a) \({f_x}\left( {1,2} \right)\) (b)\({f_y}\left( {1,2} \right)\)

Short answer

By using graph determine the signs of the partial derivatives for the function f.

(a ) \({f_x}\left( {1,2} \right)\) (b)\({f_y}\left( {1,2} \right)\)

Step-by-step solution

Step-1: Given Data:

Determine the Signs of the partial derivatives for the function f.

Step-2: Consider the following limit

Now, \(\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {1,2} \right)} \left( {5{x^3}{\rm{ - }}{x^2}{y^2}} \right)\)

The function \(f\left( {x,y} \right) = \left( {5{x^3}{\rm{ - }}{x^2}{y^2}} \right)\)is polynomial, so it is continuous at every where.

So the limits of polynomial is evaluate function at that point.

\(\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {1,2} \right)} \left( {5{x^3}{\rm{ - }}{x^2}{y^2}} \right) = f\left( {1,2} \right)\)

\(\begin{aligned}{l}f\left( {1,2} \right) &= 5 \times {\left( 1 \right)^3}{\rm{ - }}{\left( 1 \right)^2}{\left( 2 \right)^2}\\ &= 5 \times 1{\rm{ - }}1 \times 4\\ &= 5{\rm{ - 4}}\\ &= {\rm{1}}\end{aligned}\)

\(\therefore \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {1,2} \right)} \left( {5{x^3}{\rm{ - }}{x^2}{y^2}} \right) = 1\)

Hence, the answer is \(\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {1,2} \right)} \left( {5{x^3}{\rm{ - }}{x^2}{y^2}} \right) = 1\)

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