Sketch
f (x) = x + cos x on
[-2pie,2pie ]. Find any local extrema, inflection points, or asymptotes. And find the absolute maximum and absolute minimum values of f on the given interval.
i cant seem to figure out how to solve this
for local extrema f'(x) = 0
1 - sinx = 0
sinx=1
x = pi/2 or -3pi/2 for the given domain
f(pi/2) = pi/2 + cos(pi/2) = pi/2 + 0 = pi/2
for x=-3pi/2 f(-3pi/2) = -3pi/2
so the local extrema are pi/2 and -3pi/2
inflection points : f''(x) = 0
-cosx=0
x = ±pi/2 or ±3pi/2
sub those x values back into original to get the y of the points of inflection
consider the end points of the domain for absolute max/mins
f(2pi) = 2pi + cos(2pi) = 2pi + 1
f(-2pi) = -2pi + 1
compare these with the local max/mins and use your calculator to determine the largest and smallest values
BTW, no asymptotes, the graph will be a cosine curve rising as x gets larger.
To find the local extrema, inflection points, and asymptotes of the function f(x) = x + cos(x) on the interval [-2π, 2π], you need to take a few steps:
1. Start by finding the critical points of the function. Critical points occur where the derivative of the function is equal to zero or does not exist.
2. Determine the intervals where the function is increasing or decreasing by using the first derivative test.
3. Use the second derivative test to find the inflection points of the function.
4. Check for any vertical or horizontal asymptotes.
5. Finally, find the absolute maximum and minimum values of the function on the given interval.
Let's go through each step in detail:
Step 1: Finding Critical Points
To find the critical points, take the derivative of f(x):
f'(x) = 1 - sin(x)
Set it equal to zero and solve for x:
1 - sin(x) = 0
sin(x) = 1
This equation has solutions at x = π/2 + 2πn, where n is an integer.
Step 2: Determining Increasing and Decreasing Intervals
To determine where the function is increasing or decreasing, you can use the first derivative test. Check the sign of the derivative on different intervals:
Interval 1: x < π/2 + 2πn (where n = -2, -1)
In this interval, sin(x) is negative, so f'(x) = 1 - sin(x) is positive. Therefore, f(x) is increasing.
Interval 2: π/2 + 2πn < x < π/2 + 2πn+1 (where n is any integer)
In this interval, sin(x) is positive, so f'(x) = 1 - sin(x) is negative. Therefore, f(x) is decreasing.
Interval 3: π/2 + 2πn +1 < x < 2π + 2πn (where n is any integer)
In this interval, sin(x) is negative, so f'(x) = 1 - sin(x) is positive. Therefore, f(x) is increasing.
Step 3: Finding Inflection Points
To find the inflection points, you need to find where the second derivative changes sign. Take the second derivative of f(x):
f''(x) = -cos(x)
Setting f''(x) = 0, you get:
cos(x) = 0
This equation has solutions at x = π/2 + πn, where n is an integer.
Step 4: Checking for Asymptotes
To check for asymptotes, you need to examine the limit of the function as it approaches certain values.
Vertical asymptotes occur when the denominator of the function approaches zero. However, in this case, there are no values of x where the denominator becomes zero. Hence, there are no vertical asymptotes.
To check for horizontal asymptotes, evaluate the limit of the function as x approaches positive or negative infinity:
lim(x -> ±∞) (x + cos(x)) = ±∞
Since the limit diverges, there are no horizontal asymptotes.
Step 5: Finding Absolute Maxima and Minima
To find the absolute maximum and minimum values of the function on the given interval, you need to evaluate the function at the critical points and endpoints:
f(-2π) ≈ -8.283
f(-π/2) ≈ -0.845
f(π/2) ≈ 0.155
f(2π) ≈ 8.283
The absolute maximum value is approximately 8.283, which occurs at x = -2π and x = 2π since these are the endpoints of the interval. The absolute minimum value is approximately -0.845, which occurs at x = -π/2 since it is a critical point.
Now that you have completed each step, you should have the information needed to sketch the graph of f(x) = x + cos(x) on the interval [-2π, 2π].