For the function \( f(x) \), find the maximum number of real zeros, the maximum number of \( x \)-intercepts, and the maximum number of turning points that the function can have. \( f(x)=x^{7}-x^{3}+2 \)

The function \( f(x) = x^{7} - x^{3} + 2 \) is a polynomial of degree 7. This means that it can have a maximum of 7 real zeros, as per the Fundamental Theorem of Algebra. However, whether it achieves this number depends on the specific nature of the roots. For \( x \)-intercepts, which occur when \( f(x) = 0 \), we find that a polynomial of degree 7 can also have a maximum of 7 real \( x \)-intercepts. Yet, due to the nature of the function, some roots may be complex. Turning points are where the derivative changes sign, and for a degree \( n \) polynomial, the maximum number of turning points is \( n-1 \). Therefore, \( f(x) \) can have a maximum of \( 7-1 = 6 \) turning points.