Function extrema

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Problem: Examine the function for an extremum: \[ z=y \sqrt{x}-y^{2}-x+6 y . \]

2.11.1 Function extrema

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Problem: Determine the largest and smallest values of the function \( z=x^{2}-y^{2} \) in the domain \( x^{2}+y^{2} \leq 1 \).

2.11.2 Function extrema

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Problem: Determine the largest and smallest values of a function \( f(x)=2 x^{2}+\frac{2}{x} \) in \( \left[\frac{1}{2} ; 1\right] \).

2.11.3 Function extrema

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Problem: Find the points of local extrema of the function \[ z=x^{3}-y^{3}-3 x y . \]

2.11.4 Function extrema

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Problem: Find the rectangle with the largest area inscribed in a circle of radius \( R \).

2.11.5 Function extrema

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Problem: Find local extrema of the function \[ z=x^{4}+y^{4}-4 x y \text {. } \]

2.11.6 Function extrema

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Problem: Among rectangular parallelepipeds with a given volume \( V \) find the one, having the smallest total surface.

2.11.7 Function extrema

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Problem: A function of three variables is given: \( f\left(x_{1}, x_{2}, x_{3}\right) \). a) Find the stationary point of function \( f\left(x_{1}, x_{2}, x_{3}\right) \), calculate the value of the function and the gradient at it. (Indicate the method for solving the system of equations). b) Determine the extremums of function \( f\left(x_{1}, x_{2}, x_{3}\right) \), if there are any. \[ \begin{array}{l} f\left(x_{1}, x_{2}, x_{3}\right)=3 x_{1}^{2}+8 x_{2}^{2}+13 x_{3}^{2}-4 x_{1} x_{2}- \\ -8 x_{1} x_{3}-6 x_{2} x_{3}-55 x_{1}-18 x_{2}-21 x_{3} . \end{array} \]

2.11.8 Function extrema

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Problem: Function \( f\left(x_{1}, x_{2}, x_{3}\right) \) and its constraints are given. a) Compose the Lagrange function; b) Determine the stationary point of the Lagrange function; c) Determine the extremums of function \( f\left(x_{1}, x_{2}, x_{2}\right) \), if there are any. \[ \begin{array}{l} f\left(x_{1}, x_{2}, x_{3}\right)=15 x_{1}^{2}+7 x_{2}^{2}+10 x_{3}^{2}+20 x_{1} x_{2}+ \\ +x_{1} x_{3}+3 x_{2} x_{3}+10 x_{1}-25 x_{2}-20 x_{3}, \\ 11 x_{1}+20 x_{2}+10 x_{3}=300, \\ 21 x_{1}+60 x_{2}+x_{3}=270 . \end{array} \]

2.11.9 Function extrema

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