which best describes the graph of the cubic function $f(x) = x^3 + x^2 + x + 1$?\n\n- as $x$ increases, $y$…

which best describes the graph of the cubic function $f(x) = x^3 + x^2 + x + 1$?\n\n- as $x$ increases, $y$ increases along the entire graph.\n- as $x$ increases, $y$ increases, decreases, and then increases again.\n- as $x$ increases, $y$ decreases, increases, and then decreases again.\n- as $x$ increases, $y$ decreases along the entire graph.

which best describes the graph of the cubic function $f(x) = x^3 + x^2 + x + 1$?\n\n- as $x$ increases, $y$ increases along the entire graph.\n- as $x$ increases, $y$ increases, decreases, and then increases again.\n- as $x$ increases, $y$ decreases, increases, and then decreases again.\n- as $x$ increases, $y$ decreases along the entire graph.

Answer

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)