`f(x) = (x^4 + x^3 + 1)/(sqrt(x^2 + x + 1))` Estimate the intervals of concavity to one decimal place by using a computer algebra system to compute and graph `f''`.

Textbook Question

Chapter 4, 4.3 - Problem 59 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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marizi | High School Teacher | (Level 1) Associate Educator

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When using a graph of f"(x), we follow concavity test:

A a section of function f(x) is concave up when f"(x) > 0.

 This means the segment of the graph of f"(x) is above the x-axis.

A a section of function f(x) is concave down when f"(x) < 0.

 This means the segment of the graph of f"(x) is below the x-axis.

 Using computer algebra system, here is the graph of f(x) = `(x^4+x^3+1)/(sqrt(x^2+x+1))`

 

 

 For its second derivative graph:

f"(x) =

  The graph of f"(x) intersects the x-axis at x`~~` -0.7   and x`~~` 0.1.

These will be used as the boundary values to set the intervals of concavity.

These are the final answers.

Concave up: (-`oo`  ,-0.7) and (0.1, +`oo` ` ` )

Concave down: (-0.7, 0.1)

 

 In case you prefer a more accurate data, here is maximize view of the graph of f"(x).

 

  The graph of f"(x) intersects the x-axis at x=-0.64   and x=0.03.

These will be used as the boundary values to set the intervals of concavity.

Concave up: (-`oo` ,-0.64) and (0.03, +`oo` )

Concave down: (-0.64, 0.03) 

Sources:

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