Write the function: f(x) = x3 – x2 + x –
The Correct Answer and Explanation is:
To analyze the function ( f(x) = x^3 – x^2 + x ), we can investigate its key properties, including its roots, behavior, and critical points. Here’s a step-by-step breakdown:
Step 1: Finding the Roots
To find the roots of the function, we need to solve for ( f(x) = 0 ):
[
x^3 – x^2 + x = 0
]
Factoring out ( x ):
[
x(x^2 – x + 1) = 0
]
This gives us one root at ( x = 0 ). To find the other roots, we need to solve the quadratic equation ( x^2 – x + 1 = 0 ). The discriminant ( D ) of this quadratic is calculated as follows:
[
D = b^2 – 4ac = (-1)^2 – 4(1)(1) = 1 – 4 = -3
]
Since the discriminant is negative, ( x^2 – x + 1 ) has no real roots. Therefore, the only real root of the function ( f(x) ) is ( x = 0 ).
Step 2: Analyzing the Behavior
Next, we can investigate the behavior of the function by calculating its first derivative:
[
f'(x) = 3x^2 – 2x + 1
]
To find critical points, we set ( f'(x) = 0 ):
[
3x^2 – 2x + 1 = 0
]
Calculating the discriminant for this quadratic:
[
D’ = (-2)^2 – 4(3)(1) = 4 – 12 = -8
]
Again, since the discriminant is negative, there are no real critical points, indicating that ( f'(x) ) does not change sign. Since the leading coefficient of ( f'(x) ) is positive, ( f'(x) ) is always positive, meaning that the function ( f(x) ) is monotonically increasing for all ( x ).
Step 3: Conclusion
Thus, the function ( f(x) = x^3 – x^2 + x ) has one real root at ( x = 0 ) and is increasing everywhere. As ( x ) approaches negative infinity, ( f(x) ) approaches negative infinity, and as ( x ) approaches positive infinity, ( f(x) ) approaches positive infinity. This means the function crosses the x-axis at ( x = 0 ) and has no maximum or minimum points.