# Find absolute extrma of sin(cos(x))

• John O' Meara
In summary, the absolute extrema of f(x)=sin(cos(x)) on the interval [0,2pi] can occur at the endpoints of the interval or at the solutions to the equation f'(x)=0. The derivative for this function is -cos(cos(x))sin(x) and the solutions to f'(x)=0 can be found by solving for when cos(cos(x))=0, which occurs at x=pi/2, 3pi/2. The values of cos(x) can range from -1 to 1, and therefore cos(cos(x)) can take on all values in the interval [-1,1]. However, it is worth noting that cos(cos(x)) can only be positive in this interval
John O' Meara

## Homework Statement

A simple question; find the absolute extrema of f(x)=sin(cos(x)) on the interval [0,2pi].

The chain rule

## The Attempt at a Solution

Assuming that I am correct about the derivative being -cos(cos(x))sin(x), how do you solve
-cos(cos(x))sin(x)=0

Assuming you are right - when does x*y equal 0?

The absolute extrema must occur at the endpoints of the interval or at the solutions to the equation f'(x)=0. Do I change the trigonometric product into a sum using sin(A+B)+sin(A-B). When is x*y=0? when either x or y =0.

John O' Meara said:
When is x*y=0? when either x or y =0.

Can't you use this information to solve your equation?

So you are saying that sin(x)=0 => x=0,pi,2pi and how about the cos(cos(x))=0.

for what t's cos(t)=0?

For what t's cos(t)=0, t= pi/2, 3pi/2

What values can cos(x) take? If so, what values can cos(cos(x)) take?

the cos(x) can take on all values in the interval [-1,1], depending on the value of x.

Take it a step further - can you tell anything about cos([-1,1])?

And start moving on your own, my hand hurts from spoonfeeding.

I know all that, but I can't see it as a solution. Obviously cos([-1,1]) is positive only. I hope your hand doesn't hurt too much. Thanks very much for the help.

x*y=0 if y is always > 0.

## What is the function sin(cos(x))?

The function sin(cos(x)) is a combination of the sine and cosine functions. Cosine takes an input and outputs the cosine of that angle, and the sine function takes the cosine of an angle and outputs the sine of that angle. When these two functions are combined, the result is a new function that takes an input, finds the cosine of that input, and then finds the sine of the cosine value.

## What is an absolute extremum?

An absolute extremum is the highest or lowest point on a function over a given interval. It is the absolute maximum or minimum value of the function, meaning that there is no other point on the function that is higher or lower than the absolute extremum.

## How do I find the absolute extremum of a function?

To find the absolute extremum of a function, you need to take the derivative of the function and set it equal to 0. Then, solve for the x-values where the derivative is equal to 0. These x-values are the critical points of the function. Plug these critical points into the original function to find the corresponding y-values. The highest y-value will be the absolute maximum and the lowest y-value will be the absolute minimum.

## What is the domain of sin(cos(x))?

The domain of sin(cos(x)) is all real numbers. Both the sine and cosine functions have a domain of all real numbers, and when they are combined, the resulting function will also have a domain of all real numbers.

## Why is it important to find the absolute extremum of a function?

Finding the absolute extremum of a function allows us to identify the highest and lowest points on the function, which can provide valuable information about the behavior and characteristics of the function. It can also help us optimize the function for a given situation, such as finding the maximum or minimum value of a certain quantity.

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