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earnstyle [38]
2 years ago
14

Use a graphing calculator to graph f(x) = x+cos(kx). Use calculus to determine which values of k lead to a function with no loca

l maxima or minima. What conclusions can be drawn?
Mathematics
1 answer:
nataly862011 [7]2 years ago
7 0

Answer:

Below, you can see the graph of the function:

f(x) = x + cos(k*x)

for different values of k, as follows:

red: k = 1

green: k = 2

orange: k = 0.

Now let's find the values of k such that our function does not have local maxima nor local minima.

First, remember that for a given function f(x), the local maxima or minima points are related to the zeros of the first derivate of f(x).

This means that if:

f'(x0) = 0.

Then x0 is a maxima, minima or an inflection point.

Then if a function is such that the f'(x) ≠ 0 , ∀x, then this function will not have local maxima nor minima.

Now we have:

f(x) = x + cos(k*x)

then:

f'(x) = 1  - k*sin(k*x)

This function will be zero when:

1 = k*sin(k*x)

1/k = sin(k*x)

now, remember that -1 ≤ sin(θ) ≤ 1

then if 1/k is smaller than -1, or larger than 1, we will not have zeros.

And this will happen if -1 < k < 1.

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Answer:

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Step-by-step explanation:

[]Let us write an equation in slope-intercept form.

-> See attached

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-> Now our equation is y = -2x

[] The y-intercept, or b, can be found by plugging a point given to us into the equation we have so far.

-> y = -2x + b

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-> Now our equation is y = -2x + 3, which is our final answer.

Have a nice day!

     I hope this is what you are looking for, but if not - comment! I will edit and update my answer accordingly.

- Heather

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