Question

When the graph of y equal to 3cos2x achieves minimum what is the value of the y coordinate

Answer 1

In point where firs derivative = 0
function have loca minimum or maximum.

If second derivative < 0 that is local maxsimum.

If second derivative > 0 that is local minimum.

If expression 3cos2x mean :

3 cos (2 x )

then first derivative =

- 6 sin ( 2 x )

- 6 sin( 2 x ) = 0 when

sin ( 2 x ) = 0

sin theta = 0 when theta = 0

in this case 2 x = 0

when x = 0 or x = pi / 2

The period of sin x is 2 n pi

The period of sin ( 2 x ) is n pi

where n is some intefer number

So - 6 sin( 2 x ) = 0 when

x = n pi + 0 = n pi

or

x = n pi + pi / 2

or

x = n pi - pi / 2


Second derivative = - 12 cos ( 2 x )

For x = n pi second derivative < 0

for that's values of x function have maximum

For x = n pi + pi / 2 second derivative > 0

and

For x = n pi - pi / 2 second derivative > 0

for that's values of x function also have minimum

So function 3 cos (2 x ) have local minimums when

x = n pi + pi / 2

and

x = n pi - pi / 2

Answer 2

Given y = 3cos2x

you know it has amplitude of 3, so its minimum value is y = -3, since there is no y-translation.

Unless you are specifically doing calculus, I'd surely use my knowledge of trig to answer this one.

Good analysis, though, Bosnian!