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.