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Question
What is the maximum value of the function sin x + cos x?
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Solution
Let f(x) = sin x + cos x, be the interval [0, 2π]
f'(x) = cos x - sin x
For the highest and lowest values,
f'(x) = 0 ⇒ cos x - sin x = 0 ⇒ tan x = 1
`therefore x = pi/4 , (5 pi)/4`
Putting the values of x in f(x) = sin x + cos x, respectively,
At x = 0, f(0) = sin 0 + cos 0 = 1
x `= 2 pi "at," f(2 pi) = sin 2 pi + cos 2 pi = 1`
x`= pi /4 "at," f(pi/4) = sin pi/4 + cos pi/4`
`= 1/sqrt2 + 1/sqrt2 = 2/sqrt2`
`= sqrt2`
x `= (5pi)/4 "at," f((5pi)/4) `
`= sin (5 pi)/4 + cos (5 pi)/4`
`= - 1/sqrt2 + 1/sqrt2`
`= - 2/sqrt2`
`= -sqrt2`
Hence, the highest value is at x = `pi/4` = `sqrt2`.
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