# Determine the extreme values of each function on the given interval. `f(x)=2sin4x+3`, XE[0,pi]

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Expert Answers

rcmath | Certified Educator

`f'(x)=2*4*cos(4x)`

`f'(x)=0=>cos(4x)=0=>4x=Pi/2+KPi=>x=Pi/8+k/4*Pi`

Let's determine the exact values in the given interval. We will plug in value for K=0,1,2,3,4,...etc until we have values of x out of the given interval.

`x_0=Pi/8`

`x_1=pi/8+Pi/4=[3Pi]/8`

`x_2=Pi/8+Pi/2=[5Pi]/8`

`x_3=Pi/8+3/4Pi=[7Pi]/8`

`x_4=Pi/8+4/4Pi>Pi`

Thus you have the extremes at `x_0,x_1,x_2,x_3`

If you plug in the values of x in the original function, you will find that the maximum values are 5 and they are at points `x_0,x_2`

`f(Pi/8)=2sin(4*Pi/8)+3=2sin(Pi/2)+3=2+3=5`

You can repeat the same work for x2

the minimum values are 1 and they are at points `x_1, x_3`

`f(3Pi/8)=2sin(4*3*Pi/8)+3=2sin(3Pi/2)+3=-2+3=1`