2019 AMC 10A Problems/Problem 19
What is the least possible value of where is a real number?
Grouping the first and last terms and two middle terms gives , which can be simplified to . Noting that squares are nonnegative, and verifying that for some real , the answer is .
Let . Then the expression becomes .
We can now use the difference of two squares to get , and expand this to get .
Refactor this by completing the square to get , which has a minimum value of . The answer is thus .
Solution 3 (calculus)
Similar to Solution 1, grouping the first and last terms and the middle terms, we get .
Letting , we get the expression . Now, we can find the critical points of to minimize the function:
To minimize the result, we use . Hence, the minimum is , so .
Note: We could also have used the result that minimum/maximum point of a parabola occurs at .
The expression is negative when an odd number of the factors are negative. This happens when or . Plugging in or yields , which is very close to . Thus the answer is .
Solution 5 (using the answer choices)
Answer choices , , and are impossible, since can be negative (as seen when e.g. ). Plug in to see that it becomes , so round this to .
We can also see that the limit of the function is at least -1 since at the minimum, two of the numbers are less than 1, but two are between 1 and 2. I like this method
For those who want a video solution: https://www.youtube.com/watch?v=Mfa7j2BoNjI
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