Question

which best describes the graph of the cubic function $f(x) = x^3 + x^2 + x + 1$?

• as $x$ increases, $y$ increases along the entire graph.
• as $x$ increases, $y$ increases, decreases, and then increases again.
• as $x$ increases, $y$ decreases, increases, and then decreases again.
• as $x$ increases, $y$ decreases along the entire graph.

Explanation:

Step1: Analyze the derivative

To determine the behavior of the cubic function \( f(x) = x^3 + x^2 + x + 1 \), we first find its derivative. The derivative \( f'(x) \) is given by the power rule: \( f'(x)=3x^2 + 2x+ 1 \).

Step2: Analyze the discriminant of the derivative

For a quadratic function \( ax^2+bx + c \) (here \( a = 3 \), \( b=2 \), \( c = 1 \)), the discriminant is \( D=b^2-4ac \). Substituting the values, we get \( D=(2)^2-4\times3\times1=4 - 12=- 8 \). Since the discriminant \( D<0 \) and \( a = 3>0 \), the quadratic function \( f'(x)=3x^2 + 2x + 1 \) is always positive for all real values of \( x \).

Step3: Relate derivative sign to function behavior

If the derivative of a function is always positive, the function is strictly increasing on its entire domain. So, as \( x \) increases, \( y=f(x) \) increases along the entire graph.

Answer:

As \( x \) increases, \( y \) increases along the entire graph. (The first option)