Calculus Answers

Consider the function f(x)=(3x+6)/(3x+2). For this function there are two important intervals: (−∞,A) and (A,∞) where the function is not defined at A. Determine A.

A=

For each of the following intervals, tell whether f(x) is increasing (input INC ) or decreasing (type in DEC ).

(−∞,A):
(A,∞):

Note that this function has no inflection points, but we can still consider its concavity. For each of the following intervals, tell whether f(x) is concave up (input CU ) or concave down (type in CD ).

(−∞,A):
(A,∞):

Let f(x)=(4x^(2))/(x^(2)+3)
Find the point(s) at which f achieves a local maximum.

Find the point(s) at which f achieves a local minimum.

Find the interval(s) on which f is concave up.

Find the interval(s) on which f is concave down.

Find all inflection points.

Let f(x)=2x^3−24x+2

Input the interval(s) on which f is increasing.

Input the interval(s) on which f is decreasing.

Find the point(s) at which f achieves a local maximum.

Find the point(s) at which f achieves a local minimum.

Find the intervals on which f is concave up.

Find the intervals on which f is concave down.

Find all inflection points.

MULTIPLE CHOICE QUESTION

The function f has a continuous second derivative, and it satisfies f(−1)=−3, f′(−1)=0 and f′′(−1)=1.

We can conclude that
A. f has neither a local maximum nor a local minimum at -1
B. f has a local maximum at -1
C. f has a local minimum at -1
D. We cannot determine if A, B, or C hold without more information.

Let f(x)=(3x−4)/(x+6). Find the open intervals on which f is concave up (down). Then determine the x-coordinates of all inflection points of f.

1. f is concave up on the intervals
2. f is concave down on the intervals
3. The inflection points occur at x =

Let f(x)=1/(3x^(2)+1). Find the open intervals on which f is concave up (down). Then determine the x-coordinates of all inflection points of f.

Use interval notation if you are asked to find an interval or union of intervals. (This link opens instructions below)

f is concave up on the intervals

f is concave down on the intervals

The inflection points occur at x =

Let f(x)=x^2−3x−40. Find the open intervals on which f is concave up (down). Then determine the x-coordinates of all inflection points of f.

1. f is concave up on the intervals
2. f is concave down on the intervals
3. The inflection points occur at x =

If a and b are positive numbers, find the maximum value of
f(x)=x^(a)(1−x)^(b), 0≤x≤1
Your answer may depend on a and b.

maximum value =

Find the absolute maximum and minimum values of the following function over the given interval. If there are multiple points in a single category list the points in increasing order in x value and enter N in any blank that you don't need to use.

f(x)=(9cosx)/(14+7sinx), 0≤x≤2π

Absolute maxima
x = y =
x = y =
x = y =

Absolute minima
x = y =
x = y =
x = y =

Find the absolute maximum and minimum values of the following function on the given interval. If there are multiple points in a single category list the points in increasing order in x value and enter N in any blank that you don't need to use.

f(x)=8e^(−x)−8e^(−2x) , [0,1]

Absolute maxima
x = y =
x = y =
x = y =

Absolute minima
x = y =
x = y =
x = y =