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# Calculus - Critical Points and Points of Inflection

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f(x) = x^4 - 4x^3 + 10

a) Find f'(x) and f"(x).

b) Find all critical points and identify any local max/min.

c) Determine the intervals where f(x) is increasing or decreasing.

d) Find all possible points of inflections.

e) Determine the intervals where f(x) is concave up or down. Then identify any inflection point(s).

f) Use all the information above to graph the function. The following points should be clearly labeled with their coordinates:
i. x- and y-intercepts
ii. local max/min
iii. inflection point(s)

2. f(x) = 2x^2 / x^2-1

g) Find f'(x) and f"(x).

h) Find all critical points and identify any local max/min.

i) Determine the intervals where f(x) is increasing or decreasing.

j) Find all possible points of inflections.

k) Determine the intervals where f(x) is concave up or down. Then identify any inflection point(s).

l) Find all horizontal and vertical asymptotes, if any. (This applies to #2)

m) Use all the information above to graph the function. The following points should be clearly labeled with their coordinates:
i. x- and y-intercepts
ii. local max/min
iii. inflection point(s)

https://brainmass.com/math/basic-calculus/calculus-critical-points-points-inflection-246556

#### Solution Preview

The Solution file is attached.

f(x) = x^4 - 4x^3 + 10

(a) f'(x) = 4x^3 - 12x^2
f"(x) = 12x^2 - 24x

(b) f'(x) = 0  4x^3 - 12x^2 = 0
4x^2 (x - 3) = 0
x = {0, 3} are the critical ...

#### Solution Summary

Critical points and points of inflections for formatted. The functions are given for calculus. Complete, Neat and Step-by-step Solutions are provided in the attached file.

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