Question

Consider the function *f*(*x*) =
*x^*2/*x-*1 with *f '* (*x*) =
*x^*2-2*x/* (*x -* 1)^2 and *f ''*
(*x*) = 2 / (*x -* 1)^3 are given. Use these to
answer the following questions.

(a) [5 marks] Find all critical points and determine the
intervals where *f*(*x*) is increasing and where it
is decreasing, use the First Derivative Test to fifind local
extreme value if any exists.

(b) Determine the intervals where *f*(*x*) is
concave upward and where it is concave downward. Use the Second
Derivative Test to verify local extreme values from part (a).

Answer #1

consider the function f(x) = x/1-x^2
(a) Find the open intervals on which f is increasing or
decreasing. Determine any local minimum and maximum values of the
function. Hint: f'(x) = x^2+1/(x^2-1)^2.
(b) Find the open intervals on which the graph of f is concave
upward or concave downward. Determine any inflection points. Hint
f''(x) = -(2x(x^2+3))/(x^2-1)^3.

Given: f(x) = x^3 + 3x^2 - 9x + 10. (Note: x^3 means x-cubed,
and x^2 means x-squared, respectively.)
use simple words, and use mathematical equations and symbols
when and if necessary, to explain yourself
Discussed the following: the first and second derivative of
f(x); intervals where the curve is increasing and decreasing,
respectively; the critical points; the relative maximum and minimum
points; the point of inflection; where the curve is concave upward
or downward.

what does a derivative tell us?
F(x)=2x^2-5x-3, [-3,-1]
F(x)=x^2+2x-1, [0,1]
Give the intervals where the function is increasing or
decreasing?
Identify the local maxima and minima
Identify concavity and inflection points

Given the function
h(x)=e^-x^2
Find first derivative f ‘ and second derivative
f''
Find the critical Numbers and determine the intervals
where h(x) is increasing and decreasing.
Find the point of inflection (if it exists) and determine
the intervals where h(x) concaves up and concaves
down.
Find the local Max/Min (including the
y-coordinate)

f(x)= (x^2+2x-1)/x^2)
Find the
a.) x-intercept
b.) vertical and horizontal asymptote
c.) first and second derivative
d.) Is it increasing or decreasing? Identify any local
extrema
e.) Is it concave up and down? Identify any points of
reflection.

Analyze and plot the graph of f(x)= x^4/2 - 2x^3/3. for this,
find;
1) domain of f:
2)Vertical asymptotes:
3) Horizontal asymptotes:
4) Intersection in y:
5) intersection in x:
6) Critical numbers
7) intervals where f is increasing:
8) Intervals where f is decreasing:
9) Relatives extremes
Relatives minimums:
Relatives maximums:
10) Inflection points:
11) Intervals where f is concave upwards:
12) intervals where f is concave down:
13) plot the graph of f on the plane:

For the questions below, consider the following function.
f (x) = 3x^4 - 8x^3 + 6x^2
(a) Find the critical point(s) of f.
(b) Determine the intervals on which f is increasing or
decreasing.
(c) Determine the intervals on which f is concave up or concave
down.
(d) Determine whether each critical point is a local maximum, a
local minimum, or neither.

Use the second derivative to find the intervals where
f(x) = x4+8x3 is concave upward and concave
downward. Also find any points of inflection.

4. Given the function y = f(x) = 2x^3 + 3x^2 – 12x +
2
a. Find the intervals where f is increasing/f is
decreasing
b. Find the intervals where f is concave up/f is concave
down
c. Find all relative max and relative min (state which
is which and why)
d. Find all inflection points (also state
why)

- Suppose f is a function such that f′(x) = (x+ 1)(x−2)2(x−3),
so that f has the critical points x=−1,2,3. Determine the open
intervals on which f is increasing/decreasing.
- Let f be the same function as in Problem 9. Determine which,
if any, of the critical points is the location of a local extremum,
and indicate whether each extremum is a maximum or minimum.
Im confused on how to figure out if a function is increasing and
decreasing and...

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