[tex]x^2+5x+\frac{36}{x^2+5x}[/tex]
Answer (2)
Substitute u = x 2 + 5 x to simplify the expression to u + u 36 .
Apply the AM-GM inequality to find the minimum value of u + u 36 , which is 12.
Solve x 2 + 5 x = 6 and x 2 + 5 x = − 6 to find the values of x where the minimum occurs.
The minimum value of the expression is 12 .
Explanation
Understanding the Problem We are given the expression x 2 + 5 x + x 2 + 5 x 36 and we want to find its minimum value.
Substitution Let u = x 2 + 5 x . Then the expression becomes u + u 36 .
Applying AM-GM Inequality Apply the AM-GM inequality to u and u 36 : 2 u + u 36 ≥ u ⋅ u 36 .
Simplifying Simplify the inequality: 2 u + u 36 ≥ 36 = 6 Therefore, u + u 36 ≥ 12 .
Finding the Minimum The minimum value of the expression is 12, which occurs when u = u 36 , or u 2 = 36 , so u = 6 or u = − 6 .
Solving for x (Case 1) Solve for x when x 2 + 5 x = 6 : x 2 + 5 x − 6 = 0 , which factors as ( x + 6 ) ( x − 1 ) = 0 . Thus, x = − 6 or x = 1 .
Solving for x (Case 2) Solve for x when x 2 + 5 x = − 6 : x 2 + 5 x + 6 = 0 , which factors as ( x + 2 ) ( x + 3 ) = 0 . Thus, x = − 2 or x = − 3 .
Conclusion Therefore, the minimum value of the expression is 12, and it occurs at x = − 6 , − 3 , − 2 , 1 .
Examples
Consider a scenario where you want to minimize the cost of producing a certain item. Suppose the cost can be modeled by the expression x 2 + 5 x + x 2 + 5 x 36 , where x represents some production parameter. By finding the minimum value of this expression, you can determine the optimal value of x that minimizes the cost. This type of optimization problem appears in various fields, such as engineering, economics, and computer science, where minimizing a function is crucial for efficiency and cost-effectiveness.
The minimum value of the expression x 2 + 5 x + x 2 + 5 x 36 is 12, which occurs at x = − 6 and x = 1 . To find this value, we substitute u = x 2 + 5 x and apply the AM-GM inequality. This leads to the conclusion that the minimum occurs when u = 6 .
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