Question

Which of the following graphs could be the graph of the function f(x) = x4 + x3 – x2 – x?

Answer 1

For this case we have the following function:
 $x ^ 4 + x ^ 3 - x ^ 2 - x$
 We can rewrite the function to identify the zeros of it.
 When rewriting the function factoring we have: $x (x-1) (x + 1) ^ 2$
 Therefore, the zeros of the function are:
 $x = 0 x = 1 x = -1$
 Thus, the graph that contains intersections on the x axis in the points mentioned, will be the graph of the function.
 Answer:
 See attached image.

Answer 2

The given function is ⇒ f(x) = x⁴ + x³ – x² – x

To graph the function, we need to find zeros, minimum and maximum points.

The zeros of the function can be obtained as following;

f(x) = x⁴ + x³ – x² – x = (x⁴ + x³) – (x² + x) = x³ (x+1) - x (x+1) = (x+1) (x³ - x)

∴ f(x) = (x+1) x ( x² - 1) = x (x+1) (x+1) (x-1) = x (x-1) (x+1)²

So, the zeros will be at x = 0 , 1 , -1

To find the minimum and maximum points, we need to get f'(x)

∴ f'(x) = 4x³ + 3x² - 2x - 1

solve for x when f'(x) = 0 using the calculator.

∴ x = -1 , -0.39 , 0.64

Making a table between x and f(x) with taking the zeros and minimum and maximum points into considerations.

So, the graph of f(x) will be as the attached figure.