Problem

For the function $f(x)$, find the maximum number of real zeros, the maximum number of x-intercepts, and the maximum number of turring points that the function can have. \[ I(x)=x^{7}-x^{4}+4 \]

Solution

Step 1 :The function given is \(f(x) = x^{7}-x^{4}+4\).

Step 2 :The degree of the function is 7.

Step 3 :The maximum number of real zeros a function can have is determined by its degree. Therefore, this function can have at most 7 real zeros.

Step 4 :The maximum number of x-intercepts a function can have is also determined by its degree. The x-intercepts of a function are the real zeros of the function. Therefore, this function can have at most 7 x-intercepts.

Step 5 :The maximum number of turning points a function can have is one less than its degree. Therefore, this function can have at most 6 turning points.

Step 6 :Final Answer: The function \(f(x) = x^{7}-x^{4}+4\) can have at most \(\boxed{7}\) real zeros, at most \(\boxed{7}\) x-intercepts, and at most \(\boxed{6}\) turning points.

From Solvely APP
Source: https://solvelyapp.com/problems/12488/

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