# Maximum and minimum values

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Given: y = f(x) = 3x4 + 4x3

Find:

A. All critical points

B. Max - Min Values

C. Inflection points

D. Where is f(x) concave up

E. Where is f(x) concave down

F. X and Y intercepts

G. Where f(x) is increasing

H. Where f(x) is decreasing

I. Sketch the curve label

https://brainmass.com/math/optimization/maximum-minimum-values-22031

#### Solution Preview

f(x) = 3x^4 + 4x^3

=> f'(x) = 12x^3 + 12x^2 = 12x^2(x+1)

=> f''(x) = 36x^2 + 24x

=> f'''(x) = 72x + 24

=> f''''(x) = 72

A.)

For critical points:

f'(x) = 0 => 12x^2(x+1) = 0

=> x = 0, -1 --Answer

B.)

=>f''(0) = 0 : Neither max nor min

=> f'''(0) =

f''(-1) = 36 - 24 = 12: minima

f(infinity) = infinity

f(-infinity) ...

#### Solution Summary

This shows how to find critical points and concavity.

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