If x lies in first quadrant, find the maximum or minimum values of the function `4 sin x+3 cos x`.

Explanation

Let ` f(x) = 4 sin x + 3 cos x`
`rArr f'(x) = 4 cos x - 3 sin x`
For maxima or minima
` f'(x) = 0`
`rArr 4 cos x - 3 sin x =0 `
`rArr tan x 4/3`
and `f''(x) =- 4 sin x-3 cos x `
`=-[4*4/5+3*3/5]`
`=-5 lt 0`
[From eq. (1), `cos x = 3/5, sin x = 4/5`]
`:. ` The function is maximum at
and its maximum value = ` 4 sin x + 3 cos x `
`=4*4/5+3*3/5`
= 5.