Question

已知函数f(x)=-3sin^2x-4cosx+2 (1)求f(派/3）的值 （2）求f(x)的最大值和最小值

Answer 1

解：
(1)
x=π/3代入函数方程：
f(x)=-3sin²(π/3)-4cos(π/3)+2=-3(√3/2)²-4(1/2)+2=-9/4-2+2=-9/4
(2)
f(x)=-3sin²x-4cosx+2
=-3(1-cos²x)-4cosx+2
=3cos²x-4cosx-1
=3(cosx-2/3)²-7/3
当cosx=2/3时，f(x)有最小值f(x)min=-7/3
当cosx=-1时，f(x)有最大值f(x)max=6

Answer 2

f(x)=-3sin^2x-4cosx+2
=-3(1-cos^2x)-4cosx+2
=-3+3cos^2x-4cosx+2
=3cos^2x-4cosx-1
=3(cosx-2/3)^2-1-4/3
=3(cosx-2/3)^2-7/3

f(派/3）=3(1/2-2/3)^2-7/3
=3(-1/6)^2-7/3
=1/12-28/12
=-27/12
=-9/4

f(x)=3(cosx-2/3)^2-7/3
当cosx=2/3时，最小值=-7/3
当cosx=-1时，最大值=6

Answer 3

f(π/3)=-9/4-2+2=-9/4

f(x)=-3(1-cos²x)-4cosx+2
=-3cos²x-4cosx+5
=-3(cosx+2/3)²+19/3
所以cosx=-2/3，最大=19/3
cosx=1，最小=-2

Answer 4

1.f(π /3)=-3(3/4)-4*1/2+2=-9/4-2+2=-9/4
2. 记t=cosx, f(x)=-3(1-t^2)-4t+2=3t^2-4t-1=3(t-2/3)^2-7/3
t=2/3时， 最小值为-7/3
t=-1时，最大值为6