Find maximum/maximum value of `y` in the functions given below
(a) `y=5 - (x -1)^(2)` (b) `y =4x ^(2) - 4x + 7`
(c) `y= x^(3) - 3x`
`y =x^(3) - 6x^(2) + 9x + 15`
(e) `y = (sin 2x - x)`, where `- (pi)/(2) le xxle(pi)/(2)`

Answer

A, B, C, D

Explanation

(e) `y = (sin 2x -y)`
`(dy)/(dx) = 2 cos 2x - 1`
and `(d^(2)y)/(dx^(2)) =- 4 sin 2x`
Putting `(dy)/(dx) = 0`, we get
`2 cos 2x - 1 = 0`
`:. cos 2x = (1)/(2)`
or `2x = +- 60^(@) +- (pi)/(3)`
or `pi//2 le x le pi//2`
At `2x = + 60^(@), (d^(2)y)/(dx^(2))` is `-ve`, so value of `y` is
maximum. At `2x =- 60^(@), (d^(2)y)/(dx^(2))` is positive. So value of `y` is minimum.
`:. y_(max) = sin (+60^(@)) - (pi)/(6)`
`= ((sqrt(3))/(2)-(pi)/(6))` at `2x = (pi)/(3)`
or `x = pi//6`
and `y_(mix) = sin (-60^(@)) +(pi)/(6)`
at `x = - (pi)/(6)`
`= ((pi)/(6)-(sqrt(3))/(2))`