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Question
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
f (x) = sin x + cos x , x ∈ [0, π]
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Solution
Given function f(x) = sin x + cos x, on the interval [0, `pi`]
f'(x) = cos x - sin x
If f'(x) = 0
Then cos x - sin x = 0
`therefore tan x = 1 => x = pi/4`
At x = 0, f(0) = sin 0° + cos 0° = 1
At `x = pi/4, f(pi/4) = sin pi/4 + cos pi/4`
`= 1/sqrt2 + 1/sqrt2`
`= 2/sqrt2`
`= sqrt2`
At x = π, f (π) = sin π + cos π
= 0 - 1
= - 1
`therefore` absolute highest value = `sqrt2`
and absolute minimum value = -1
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