Which best describes the graph of the cubic function f(x) = x3 + x2 + 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.

As x increases, y increases along the entire graph.

Explanation:

1) That function can be factored in this way:

a) group terms: (x³ + x²) + (x + 1)

b) take common factor x² for the terms of the first parenthesis: x² (x + 1) + (x + 1)

c) take common factor x + 1: (x + 1) (x² + 1).

d) From that you can find the x-intercept doing:

(x + 1) (x² + 1) = 0  ⇒ one of the two factors is equal to 0, i.e.

x + 1 = 0 or x² + 1 = 0

x + 1 = 0 ⇒ x = - 1, while x² + 1 is never 0.

2) Therefore, the only x -intercept is x = - 1.

Which will help you to graph the function.

3)The increasing functions have positive slope, whilce decreasing functions have negative slope.

The slope of a function is its derivative.

The derivative of the function                 is:

That is a quadratic function (parabola) that is always positive (or 0).

It is always positive because it has not real solutions (the determinant is negative,  b² - 4ac = 2² - 4(3)(1) = 4 - 12 = - 8), and probing any value, like x = 1, the function is positive.

Therefore you have shown that the function is increasing in any interval.

For better visualization of the function see the attached the graph which shows that it is an increasing function.

It would be the first answer choice. y increases along the entire graph.

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