- #1

Mr Davis 97

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I am trying to find the max and min values of the function ##f(x,y) = 2\sin x \sin y + 3\sin x \cos y + 6 \cos x##. By the Cauchy-Schwarz inequality, we have that ##|f(x,y)|^2 \le (4+9+36) (\sin^2 x \sin^2y + \sin^2 x \cos^2 y + \cos^2 x) = 49##. Hence ##-7 \le f(x,y) \le 7##.

My question has to do with the last inequality. What information does this inequality convey exactly? Does it say that f(x,y) actually takes on all values and only the values in the interval ##[-7,7]##,and hence -7 and 7 are the min and max? Or does it say -7 and 7 are bounds on the range of f(x,y), and the max and min could actually be smaller values contained in the interval?

My question has to do with the last inequality. What information does this inequality convey exactly? Does it say that f(x,y) actually takes on all values and only the values in the interval ##[-7,7]##,and hence -7 and 7 are the min and max? Or does it say -7 and 7 are bounds on the range of f(x,y), and the max and min could actually be smaller values contained in the interval?

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