Checkmark type function

Foreword

Checkmark type function \ (f (x) = x + \ cfrac {1} {x} \) is a high school math is very special and high-frequency test function, which is a more general case \ (f (x) = \) \ (AX \) \ (+ \ cfrac {X} {B} \) \ ((A, B> 0) \) , because it looks like an image tick \ (\ Checkmark \) , so many people image It is called "check mark function", or another called "Nike function."

Model interpretation

Example 1 Judging function by the derivative method \ (y = x + \ cfrac {1} {x} \) monotonicity and for its range.

Analysis: The domain of the function is \ (the X-\ in (- \ infty, 0) \ Cup (0, + \ infty) \) , and is an odd function,

Therefore, only the first study \ (x \ in (0, + \ infty) \) image, the research tool is the derivative.

First, we get the \ (F '(X) = l- \ cfrac. 1} {2} {X ^ = \ cfrac X {2-1} ^ {X} ^ 2 \) ,

Order \ (F '(X)> 0 \) , i.e. \ (X ^ 2-1> 0 \) , to give \ (X>. 1 \) ;

Order \ (F '(X) <0 \) , i.e. \ (X ^ 2-1 <0 \) , to give \ (0 <X <. 1 \) ; binding properties odd function,

Found, function in the interval \ ((- \ infty, -1 ] \) monocytogenes, in \ ([- 1, 0) \) single down, in \ ((0,1] \) single down, in the section \ ([1, + \ infty) \) monocytogenes,

And \ (F (. 1) = 2, F (-1) = - 2 \) , made schematic function,

Known function of range \ ((- \ infty, -2] \ Cup [2, + \ infty) \) .

Another solution: \ (| Y | = | X + \ cfrac {. 1} {X} | = | X | + | \ cfrac {. 1} {X} | \ geqslant 2 \) , i.e. \ (| y | \ geqslant 2 \) ,

Therefore, as a function of range \ ((- \ infty, -2] \ Cup [2, + \ infty) \) .

Example 2 Judging function by the derivative method \ (y = 3x + \ cfrac {2} {x} \) monotonicity and for its range.

Practical application

Typical Example Analysis