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Let a and b be positive real numbers. If x^2 + y^2<=1 then what is the largest that ax+by can get?

By the Cauchy-Schwartz inequality, we have (x² + y²)(a²+b²) >= (ax+by)².
This can be transformed into (ax+by)² <= (x² + y²)(a²+b²) <= 1 * (a²+b²) <= (a²+b²). Hence ax + by <= sqrt(a^2 + b^2) and the equality is achieved when there exists ay = bx.

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Let a and b be positive real numbers. If x^2 + y^2<=1 then what is the largest that ax+by can get?

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