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Question
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
`h(x) = sinx + cosx, 0 < x < pi/2`
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Solution
given function, `h(x) = sin x + cos x, 0 < x < pi/2`
= h' (x) = cos x - sin x for all `x in (0, pi/2)`
For critical points, let h'(x) = 0
= cos x - sinx = 0
∴ h'(x) = cos x - sin x = cos x (1 - tan x)
`= tan x = 1 = x pi/4`
At x = `pi/4`, if the value of x is kept a little less than `pi/4`, then tan x will be less than 1 and if the value of x is kept a little more than `pi/4`, then tan x will be more than 1.
Thus, the sign of 1 - tan x changes from positive to negative and there is no change in sign in cos x.
Hence, x = `pi/4,`h is maximum.
Local maximum value = h `= (pi/4) = sin pi/4 + cos pi/4`
`= 1/sqrt2 + 1/sqrt2`
`= 2/sqrt2`
`= sqrt2`
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