If c is a zero of even multiplicity for the function f, the graph of f touches the x-axis at c but does not cross it.
This behavior arises because a zero of even multiplicity means that the function has a factor of the form
(x – c)n where n is an even number. Consequently, when f is evaluated at c, the terms related to the (x – c) factor vanish, resulting in a zero value. However, as you approach c from either side, the values of f will remain non-negative or non-positive, depending on the leading coefficient of the polynomial.
This means that near c, the graph will resemble a U-shape (like a parabola opening upwards if the leading coefficient is positive), which visually represents the graph touching the axis rather than crossing it.