Prove that a function does not have a fixed point

[Amer]

it is a question in my book said

Prove that the function $$f(x) = 2 + x - \tan ^{-1} x$$ has the property $$\mid f'(x)\mid < 1$$
Prove that f dose not have a fixed point

but i found that this function has a fixed point

$$y = 2 + y - \tan ^{-1} y$$

$$y = \tan 2$$

is it right that the question is wrong

 

[Evgeny.Makarov]

Re: Prove that a function dose not have a fixed point

Hint: $\tan^{-1}(\tan 2)\ne 2$.

 

[Amer]

Re: Prove that a function dose not have a fixed point

Hint: $\tan^{-1}(\tan 2)\ne 2$.

can you explain more please

 

[Evgeny.Makarov]

Re: Prove that a function dose not have a fixed point

Since tan(x) is periodic, for each y (such as y = tan(2)) there exists an infinite number of x such that tan(x) = y. Therefore, a convention is needed to select a single value for $\tan^{-1}(y)$. By definition, arctangent returns values in the interval $(-\pi/2,\pi/2)$, called principal values. Therefore, $\tan^{-1}(\tan(2))=2-\pi$.

 

[blue_raver22]

The question is not necessarily wrong, but it may be incomplete or misleading. It is important to fully understand the properties and assumptions of a function before attempting to prove or disprove any statements about it.

In this case, the given function f(x) = 2 + x - \tan ^{-1} x does have a fixed point at y = \tan 2, as you have correctly shown. However, the statement that \mid f'(x)\mid < 1 does not necessarily mean that the function does not have a fixed point.

The statement \mid f'(x)\mid < 1 means that the absolute value of the derivative of the function is always less than 1. This is a necessary but not sufficient condition for a function to have a fixed point. In other words, if a function does not have this property, then it definitely does not have a fixed point. However, just because a function does have this property, it does not guarantee that it will have a fixed point.

To prove that a function does not have a fixed point, we need to show that there is no value of x that satisfies the equation f(x) = x. In this case, we can see that y = \tan 2 is the only value that satisfies this equation, so the function does have a fixed point. Therefore, the original statement is incorrect.

In conclusion, it is important to carefully consider the properties and assumptions of a function before attempting to prove or disprove any statements about it. In this case, the function does have a fixed point and the given statement does not accurately reflect this fact.