Let x and y be real numbers such that x^2 + y^2 = 1. Find the max value of 2x - 5y.
A hint was given to use Cauchy-Schwarz but I'm not sure how to apply it to vectors?
To solve this problem using Cauchy-Schwarz inequality, let's consider the following vectors:
u = (x, y)
v = (2, -5)
By applying Cauchy-Schwarz inequality, we have:
|(u · v)| ≤ ||u|| ||v||
where u · v represents the dot product of vectors u and v, and ||u|| and ||v|| represent the lengths (or magnitudes) of u and v, respectively.
The dot product of two vectors is given by:
u · v = (x)(2) + (y)(-5) = 2x - 5y
The length (or magnitude) of a vector u = (x, y) is given by:
||u|| = √(x^2 + y^2)
Substituting these values into the Cauchy-Schwarz inequality, we have:
|2x - 5y| ≤ √[(x^2 + y^2)(2^2 + (-5)^2)]
Simplifying further, we get:
|2x - 5y| ≤ √[(x^2 + y^2)(4 + 25)]
|2x - 5y| ≤ √[(x^2 + y^2)(29)]
|2x - 5y| ≤ √(29)
Now, since we know that x^2 + y^2 = 1 (from the given condition), we can substitute this value into the inequality:
|2x - 5y| ≤ √(29)
Finally, the maximum value of 2x - 5y will occur when the absolute value of 2x - 5y is equal to √(29). So, the maximum value is √(29).
Therefore, the max value of 2x - 5y is √(29)