Please help. show work!! 1) For the function F(x) = x + sin (2x) on [0, pi] Find
ID: 3341356 • Letter: P
Question
Please help. show work!!
1) For the function F(x) = x + sin (2x) on [0, pi] Find all intervals of increase and decrease.
2) For the function F(x) = sin^2x - 2 cos x on [0, pi] Find all intervals of increase and decrease
3) For the function F(x) = x^4 - 2x^2 + 3 on [-2,3] Find all intervals of increase and decrease
4) For the function F(x) = x / x^2 + 1 on [0,2] Find all intervals of increase and decrease
5) For the function F(x) = x^3 - 27x - 20 on [-3,5] Find all intervals of increase and decrease
Explanation / Answer
1) f'(x) = 1 + 2cos(2x)
For increase,
f'(x) >= 0 => (1+2cos(2x) ) >= 0 => cos(2x) >= -1/2 => x = [0,pi/3] U [2pi/3,pi]
For decrease,
f ' (x) <= 0 => x = [pi/3, 2pi/3]
2) f ' = 2sinxcosx + 2sinx = 2sinx(cosx+1)
For increase, f ' >= 0
=> 2sinx(cosx + 1) >= 0 => x = [0, pi]
For decrease, f ' <= 0
f does not decrease in [0, pi]
3) f ' = 4x^3 - 4x = 4x(x-1)(x+1)
For increase, f ' >= 0
=> 4x(x-1)(x+1) >= 0 => x = [-1, 0] U [1, 3]
For decrease, f ' <= 0
=> x = [-2, -1] U [0, 1]
4) f ' = ( 1 - x^2 ) / (x^2 + 1)^2 = (1-x)(1+x) / (x^2 + 1)^2
For increase, f ' >= 0
=> (1-x)(1+x) / (x^2 + 1)^2 >= 0 => x = [0, 1]
For decrease, f ' <= 0
=> x = [1, 2]
5) f ' = 3x^2 - 27 = 3(x-3)(x+3)
For increase f ' >= 0
=> 3(x-3)(x+3) >= 0 => x = [3, 5]
For decrease, f ' <= 0
=> x = [-3, 3]