If `A=4sintheta+cos^2theta,` then which of the following is not true? (a) maximum value of `A is 5.` (b)minimum value of `A is -4` (c) maximum value of `A` occurs when `sintheta=1/2dot` (d) Minimum value of `A` occurs when `sin theta`=1

Options

A. Maximum value of A is 5 .
B. Minimum value of A is `-4`
C. Maximum value of A occurs when `sin theta = 1//2`
D. Minimum value of A occurs when `sin theta = 1`.

Answer

ACD

Explanation

(a) , ( c) , (d)
`f(theta) = 4 sin theta cos^(2) theta = 4 sin theta + 1 - sin^(2) theta`
`= 5 -(4-4sin theta + sin^(2) theta) = 5 - (sin theta -2)^(2)`
Now maximum value of f`(theta)` occurs when `(sin theta -2)^(2)` is minimum .
Minimum value of `(sin theta -2)^(2)` occurs when `sin theta =1` , then maximum value of `f(theta)" is " 5-(-1-2)^2=4`.
Also minimum value of `f(theta)` occurs when `(sintheta-2)^2` is maximum.
Mximum value of `(sintheta-2)^2` occurs when `sintheta=-1`, then minimum value of `f(theta)" is " 5-(-1-2)^2=-4`.