a^4 -16a -12

Input:

a4 - 16 a - 12 Plots: 10 1

2

-10

3

Ha from -0.5 to 2.5L

-20 -30

Alternate form:

Ia2 - 2 a - 2M Ia2 + 2 a + 6M Real roots:

a ‡ 1-

3

Complex roots:

a ‡ -1 - ä

5

Generated by Wolfram|Alpha (www.wolframalpha.com) on November 15, 2009 from Champaign, IL. © Wolfram Alpha LLC—A Wolfram Research Company

1

a^4 -16a -12

Roots in the complex plane: Im

Re

Polynomial discriminant:

D ‡ -2 211 840 Derivative:

â âa

Ia4 - 16 a - 12M ‡ 4 a3 - 16

Indefinite integral: 4 à Ia - 16 a - 12M â a ‡

a5 5

- 8 a2 - 12 a + constant

Global minimum:

min 9a4 - 16 a - 12= ‡ -12 - 12 ´ 223 at a ‡ 223 Definite integral area below the axis between the smallest and largest real roots: 1+ 3

à

I-12 - 16 a + a4 M ΘI12 + 16 a - a4 M â a ‡ -

1- 3

192

» -66.5108 5

Generated by Wolfram|Alpha (www.wolframalpha.com) on November 15, 2009 from Champaign, IL. © Wolfram Alpha LLC—A Wolfram Research Company

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2