k∈R,求(4k+3)/(1+k²)的取值范围

k∈R,求(4k+3)/(1+k²)的取值范围
k∈R,求(4k+3)/(1+k²)的取值范围

k∈R,求(4k+3)/(1+k²)的取值范围
令a=(4k+3)/(1+k²)
4k+3=a+ak²
ak²-4k+(a-3)=0
k是实数则方程有实数解
所以△>=0
16-4a²+12a>=0
a²-3a-4=(a+1)(a-4)

let
f(x) = (4x+3)/(1+x^2)
f'(x) = [4(1+x^2)- (4x+3)(2x) ]/(1+x^2)^2=0
4(1+x^2)- (4x+3)(2x) =0
2x^2+3x-2=0
(2x-1)(x+2)=0
x=1/2 (max) or -2(min)
f(1/2) = ( 4(1/2)+3) /(1+(1/2...

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let
f(x) = (4x+3)/(1+x^2)
f'(x) = [4(1+x^2)- (4x+3)(2x) ]/(1+x^2)^2=0
4(1+x^2)- (4x+3)(2x) =0
2x^2+3x-2=0
(2x-1)(x+2)=0
x=1/2 (max) or -2(min)
f(1/2) = ( 4(1/2)+3) /(1+(1/2)^2)
= 4
f(-2) = -1
取值范围 =[-1,4]

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