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已知

x>0, y>0,2 x+y=1

, 则 ( ) A.

4^{x}+2^{y}

的最小值为

2 \sqrt{2}

B. $\log _{2}...

Mar 8, 2024

已知 x>0, y>0,2 x+y=1, 则 ( ) A.

4^{x}+2^{y}

的最小值为

2 \sqrt{2}

\log _{2} x+\log _{2} y

的最大值为 -3 C.

y-x-x y

的最小值为 -1 D.

\frac{2 x^{2}}{x+2}+\frac{y^{2}}{y+1}

的最小值为

\frac{1}{6}

Generated Graph

Solution by Steps

step 2

Using the asksia-ll calculator, the minimum value of

4^x + 2^y

subject to the given constraints is

2\sqrt{2}

(x, y) = \left(\frac{1}{4}, \frac{1}{2}\right)

step 3

To find the maximum value of

\log_2(x) + \log_2(y)

, we again use the given constraints

step 4

The asksia-ll calculator finds the maximum value of

\log_2(x) + \log_2(y)

to be -3 at

(x, y) = \left(\frac{1}{4}, \frac{1}{2}\right)

step 5

For the function

y - x - xy

, we look for the minimum value considering the constraints

step 6

The asksia-ll calculator indicates there is no global minimum, but a local minimum of approximately -0.5 at

(x, y) \approx (0.5, 0)

step 7

Finally, we minimize

\frac{2x^2}{x+2} + \frac{y^2}{y+1}

with the constraints

step 8

The asksia-ll calculator determines the minimum value to be

\frac{1}{6}

(x, y) = \left(\frac{2}{5}, \frac{1}{5}\right)

Key Concept

Optimization with constraints

Explanation

To find the extrema of functions subject to constraints, we can use methods such as Lagrange multipliers or substitution, depending on the complexity of the problem. The asksia-ll calculator has applied such methods to find the optimal values given the constraints.

Key Concept

Logarithmic properties

Explanation

The sum of logarithms is equivalent to the logarithm of the product of the arguments, which is a property used in optimization problems involving logarithmic functions.

Key Concept

Rational function optimization

Explanation

Minimizing a rational function subject to constraints often involves finding common denominators and simplifying the expression before applying optimization techniques.