Question

Which statement describes the graph of this polynomial function? f (x) = x Superscript 4 Baseline + x cubed minus 2 x squared The graph crosses the x-axis at x = 2 and x = –1 and touches the x-axis at x = 0. The graph touches the x-axis at x = 2 and x = –1 and crosses the x-axis at x = 0. The graph crosses the x-axis at x = –2 and x = 1 and touches the x-axis at x = 0. The graph touches the x-axis at x = –2 and x = 1 and crosses the x-axis at x = 0.

Answer 1

The correct answer is C, or The graph crosses the x-axis at x = –2 and x = 1 and touches the x-axis at x = 0.

Just got it right on edge 2020, hope this helps and good luck! :)

Answer 2

The graph crosses the x-axis at x = –2 and x = 1 and touches the x-axis at x = 0.

Explanation:

Given, f(x) = x⁴ + x³- 2 x².

Now, f(x) to touch or cross x-axis, f(x) must be equal to 0.

⇒ f(x) =0 ; x⁴ + x³- 2 x² = 0;

⇒ x²(x²+x-2) = 0

⇒ x=0 or x=-2 or x=1 .

now, for f(x) to touch x-axis, f'(x) = 0 at these three points(x=0,-2,1). where f'(x) is first derivative of x.

as f'(x) will be 0 if local maximum or minimum exists, thus touches the axis.

Now, f'(x) = 4 x³+ 3 x² - 4 x;

and only when x=0; f'(x) = 0.

⇒ The graph crosses the x-axis at x = –2 and x = 1 and touches the x-axis at x = 0.