# Calculus - Critical Points and Points of Inflection

Please see the attached file for the fully formatted problems.

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)

#### Solution Preview

...0) = 0 and f"(3) = 36

f(0) = 10 and f(3) = -17

(0, 10) is neither a point of maximum nor a point of minimum. It is a point of inflection

(3, -17) is a point of local minimum

(c) f'(x) > 0 ...