Increasing and Decreasing Functions (max/min)

In summary, the problem is to find a value for a so that the curves y = sin ax and y = cos ax intersect at right angles at the point (x0, y0). The solution involves using the unit circle and the fact that the multiplication of the slopes of two perpendicular lines is -1. The first step is to set up an equation using the derivatives of the two functions, and then solve for a.
  • #1
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Homework Statement



Find a > 0 so that the curves y = sin ax and y = cos ax intersect at right angles (let them intersect at (x0, y0)).


2. The attempt at a solution

Thinking about the unit circle, if theta equals pi/4, then sin theta and cos theta would intersect at right angles at the point (1/sqrt2, 0). Does this imply that sin ax0 = cos ax0? I don't know where to go from here. This problem is at the end of a section that concerned what I can learn about the graph of a function from the first derivative of the function. However, I don't see how the derivatives of the above functions can help me here. I would greatly appreciate a walk-through.
 
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  • #2
you know one thing if they intersect at right angle, the multiplication of their slope should be negative 1.
say
m1=3
m2=-1/3
that means that they are perpendicular

Now, what do derivatives give you ?

I'm not sure if this works, but you should try
 

1. What is an increasing function?

An increasing function is a mathematical function that has a positive slope, meaning that as the input value increases, the output value also increases. This can be represented graphically as a line that slopes upwards from left to right.

2. How do you determine if a function is increasing or decreasing?

To determine if a function is increasing or decreasing, you can look at the slope of the function. If the slope is positive, the function is increasing. If the slope is negative, the function is decreasing. You can also look at the graph of the function and see if it is going up or down from left to right.

3. How do you find the maximum or minimum value of a function?

To find the maximum or minimum value of a function, you can take the derivative of the function and set it equal to 0. Then, solve for the input value that makes the derivative equal to 0. This input value will correspond to the maximum or minimum value of the function. You can also look at the graph of the function and see the highest or lowest point, which will be the maximum or minimum value.

4. What is the difference between a local maximum/minimum and a global maximum/minimum?

A local maximum/minimum is the highest/lowest point in a specific region of a function. It may not be the absolute highest/lowest point of the entire function. A global maximum/minimum, on the other hand, is the absolute highest/lowest point of the entire function. It is the highest/lowest point out of all possible input values.

5. Can a function have more than one maximum or minimum?

Yes, a function can have more than one maximum or minimum. This can happen when the function has multiple peaks or valleys. These points are called local maximums or minimums, while the absolute highest or lowest point is the global maximum or minimum.

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