Question

which statement describes the graph of $f(x)=-x^{4}+3x^{3}+10x^{2}$? the graph crosses the x - axis at $x = 0$ and touches the x - axis at $x = 5$ and $x=-2$. the graph touches the x - axis at $x = 0$ and crosses the x - axis at $x = 5$ and $x=-2$. the graph touches the x - axis at $x = 0$ and crosses the x - axis at $x=-5$ and $x = 2$. the graph crosses the x - axis at $x = 0$ and touches the x - axis at $x=-5$ and $x = 2$.

Explanation:

Step1: Factor the function

$f(x)=-x^{4}+3x^{3}+10x^{2}=-x^{2}(x^{2}-3x - 10)=-x^{2}(x - 5)(x+2)$

Step2: Analyze the roots and their multiplicities

The roots of the function are found by setting $f(x) = 0$. So $-x^{2}(x - 5)(x + 2)=0$. The roots are $x = 0$, $x=5$ and $x=-2$. The root $x = 0$ has multiplicity 2 (since the factor is $x^{2}$), and the roots $x = 5$ and $x=-2$ have multiplicity 1.
When the multiplicity of a root is even, the graph touches the $x$-axis at that root. When the multiplicity is odd, the graph crosses the $x$-axis at that root. So the graph touches the $x$-axis at $x = 0$ and crosses the $x$-axis at $x=5$ and $x=-2$.

Answer:

The graph touches the $x$-axis at $x = 0$ and crosses the $x$-axis at $x = 5$ and $x=-2$.