Page 60 - Calc for the Clueless

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For y = x - 3x , where is the function increasing and decreasing? What about its concavity?
y increases if the slope of y' is positive. Look at any graph that increases.

Solving this quadratic inequality, we find that the curve increases if x > 2 or x < 0. That would mean y
decreases when 0 < x < 2.

Before, we found that a curve faces up if y'' > 0. y" = 6x - 6 > 0 if x > 1, which is where the curve is up, and y"
< 0 if x < 1, where the curve is down.

You might try to do a sketch of this one on your own. If so, don't look at what follows.

Intercepts: (0,0), (3,0). Max, min: (0,0) is the max, (2,-4) is the min. Inflection point: (1,-2). Right end to plus
infinity and left end to minus infinity. Polynomials have no asymptotes. And the sketch is...

Answers

Answers to the problem on page 68:

x A B C D E F G H I J K

f(x) - - + 0 ? - 0 + + + +

f'(x) + ? 0 - ? + + - 0 ? -

f"(x) 0 ? - - ? 0 + + + ? 0