Max/min with trig funtions problem

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In summary, the question asks for the minimum and maximum values of f(x) within the given interval. To find these values, the first derivative is set to zero and solved for x. However, this only gives one solution, x = -6.56487, which is the minimum value. The period of tan(0.1 x) is needed to find the maximum value, as there will be another solution within the given interval.
  • #1
icosane
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Homework Statement



Within the interval ( -22.27, 40.55 ), for what value of x does f(x) take on a minimum if
f(x)=4.7sin(0.1x)−6.1cos(0.1x)

Later the question asks for the maximum as well.

2. The attempt at a solution

I know to find the critical values to take the first derivative and set it equal to zero, then solve for x. When I do that I get the equation...

0=.47cos(.1x)+.61sin(.1x)

When I solve this guy, by manipulating the equation to get tan(.1x)=-.47/.61, I get only one value, x=-6.56487, which is the minimum value. But I also need a maximum value. I've tried both end points within the interval and they were wrong. Is there something I'm missing here?

Any help would be appreciated.
 
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  • #2
The tangent is a periodic function.
What is the period of tan(0.1 x)?
One period beyond x = -6.5... there will be another solution which is also in the given interval.
 

1. What is a "Max/min with trig functions problem"?

A "Max/min with trig functions problem" is a type of mathematical problem that involves finding the maximum or minimum value of a trigonometric function within a given domain. This is typically done by finding the critical points of the function and evaluating them to determine the maximum or minimum value.

2. How do I identify the critical points in a "Max/min with trig functions problem"?

In order to identify the critical points in a "Max/min with trig functions problem", you must first take the derivative of the given trigonometric function. The critical points will be the values of x where the derivative is equal to 0 or undefined. These points will then need to be evaluated to determine the maximum or minimum value.

3. Can I use the unit circle to solve a "Max/min with trig functions problem"?

Yes, the unit circle can be a useful tool in solving "Max/min with trig functions problems" because it helps to visualize the behavior of the trigonometric function within a given domain. By identifying the coordinates of the critical points on the unit circle, you can determine the maximum or minimum value of the function.

4. Are there any special rules for solving "Max/min with trig functions problems"?

Yes, there are a few special rules that can be helpful in solving "Max/min with trig functions problems". These include the fact that the maximum or minimum value of a trigonometric function will occur at either the endpoints of the given domain or at a critical point. Additionally, the maximum or minimum value may also occur at a point where the function is undefined.

5. How can I check my solution to a "Max/min with trig functions problem"?

To check your solution to a "Max/min with trig functions problem", you can use a graphing calculator or software to graph the function and visually verify that the critical points you found are indeed the maximum or minimum values. Additionally, you can plug in the critical points and endpoints into the original function to ensure that they produce the correct maximum or minimum value.

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