(1) construction method (including structural vector method, structural complex method, structural diagram method, etc. )
[Example] Find the range of f(x)=√(2-x)+√(x- 1).
The construction vector m=( 1, 1), n=(√(2-x), √(x- 1)),
Then according to the vector norm inequality | m ||| n | ≥| m n |.
( 1? + 1? )[(2-x)+(x- 1)]≥[ 1 √( 2-x)+ 1 √( x- 1)]？
→0 & lt； √(2-x)+√(x- 1)≤√2。
Therefore, the function value domain is (0, √2).
(2) Discrimination method
[Example] Find y=x? +4x+9 range.
The original formula is x? +4x+(9-y)=0。
The above discriminant is not less than 0, that is
16-4(9-y)≥0
Solution, y≥5.
That is, the function range is [5, +∞).
(3) Inequality method:
[Example] Find the range of trigonometric function f(x)= 1/sinx+8/cosx in the definition domain (0, π/2).
According to weight and inequality
f(x)= 1/sinx+8/cosx
= 1^(3/2)/(sin？ x)^( 1/2)+4^(3/2)/(cos？ x)^( 1/2)
≥( 1+4)^(3/2)/(sin？ x+cos？ x)^( 1/2)
=5√5,
So the function value domain is [5√5, +∞).
(4) Derivative method
[Example] X > 0, find the range of f(x)=(㏑x)/x 。
The derivative of the original formula is f ′ (x) = (1-㏑ x)/x? .
f′(x)>0, 0
When f'(x)=0, x = e；;
f′(x)& lt； 0，x & gte.
∴f(x)|max=f(e)= 1/e.
So the function value domain is (-∞, 1/e).
(5) Linear programming method
[Example] (omitted)
(6) Monotonicity method of function
[Example] (omitted)
There are many methods, so I won't list them one by one!