Question Suppose that $$f(x, y)=x^{2}+y^{2}$$ at which $$0 \leq x, y$$ and $$3 x+9 y \leq 8$$. Please enter exact answers. 1. Absolute minimum of $$f(x, y)$$ is 2. Absolute maximum of $$f(x, y)$$ is

Transcribed Image Text: Suppose that $$f(x, y)=x^{2}+y^{2}$$ at which $$0 \leq x, y$$ and $$3 x+9 y \leq 8$$. Please enter exact answers. 1. Absolute minimum of $$f(x, y)$$ is 2. Absolute maximum of $$f(x, y)$$ is
Transcribed Image Text: Suppose that $$f(x, y)=x^{2}+y^{2}$$ at which $$0 \leq x, y$$ and $$3 x+9 y \leq 8$$. Please enter exact answers. 1. Absolute minimum of $$f(x, y)$$ is 2. Absolute maximum of $$f(x, y)$$ is
&#12304;General guidance&#12305;The answer provided below has been developed in a clear step by step manner.Step1/3given \begin{align*} \mathrm{{f{{\left({x},{y}\right)}}}} &= \mathrm{{x}^{{2}}+{y}^{{2}}} \end{align*}at which , \begin{align*} \mathrm{{0}} &\le \mathrm{{x},{y}} \end{align*}and \begin{align*} \mathrm{{3}{x}+{9}{y}} &\le \mathrm{{8}} \end{align*}first obtaining critical points of function by putting :$$\mathrm{{{f}_{{{x}}}}}$$ = 0 and $$\mathrm{{{f}_{{{y}}}}}$$ = 0 $$\mathrm{{{f}_{{{x}}}}}$$ = 2x , $$\mathrm{{{f}_{{{y}}}}}$$ = 2ythus critical point is : (0,0)$$\mathrm{{{f}_{{{x}{x}}}}}$$ = 2 , $$\mathrm{{{f}_{{{y}{y}}}}}$$ = 2 , $$\mathrm{{{f}_{{{x}{y}}}}}$$ = 0 , D = (2)(2) - 0 = 4 &gt; 0,hence absolute minimum will be obtained at (0,0)putting it into the function we get ,absolute minimum = f(0,0) = 0 Explanation:as we have obtained D(x,y) &gt; 0 hence obtained critical point will be absolute minimumStep2/3f(x,y) = x^2 + y^2checking the boundary : g(x,y) = 3x + 9y = 8using method of LaGrange's multiplier :\begin{align*} \mathrm{{{f}_{{{x}}}}} &= \mathrm{\lambda{{g}_{{{x}}}}} \end{align*}2x = 3$$\mathrm{\lambda}$$x = \( \mathrm{\frac{{{3}\lambd ... See the full answer