Answer:
The graph crosses the x-axis at x = –2 and x = 1 and touches the x-axis at x = 0.
Step-by-step 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.
