Finding Minimum and Maximum values.

[bah83]

Homework Statement

Find the minimum and maximum values of x and the values of f(x) at these points if:
f(x) = 10.9sin(0.7x)+2.1cos(0.7x)
The range is (-.27,8.7)

Homework Equations

I don't know/think there are any

The Attempt at a Solution

I found f'(x) = 7.43cos(0.7x)-1.47sin(0.7x) = 0 and simplified to tan(0.7x) = 7.43/1.47. so arctan(7.43/1.47) = 0.7x or so I thought, but that doesn't seem to be the case as I can't find a value of x within the range.
Please give me some kind of advice, any help is welcome, thank you in advance.

 

[Ray Vickson]

Homework Statement

Find the minimum and maximum values of x and the values of f(x) at these points if:
f(x) = 10.9sin(0.7x)+2.1cos(0.7x)
The range is (-.27,8.7)

Homework Equations

I don't know/think there are any

The Attempt at a Solution

I found f'(x) = 7.43cos(0.7x)-1.47sin(0.7x) = 0 and simplified to tan(0.7x) = 7.43/1.47. so arctan(7.43/1.47) = 0.7x or so I thought, but that doesn't seem to be the case as I can't find a value of x within the range.
Please give me some kind of advice, any help is welcome, thank you in advance.

Try plotting a graph of the function over the range of x; this will give you some insights. In fact: if you can do it easily (eg., using some convenient computer package) graphing should almost always be your first step.

 

[bah83]

I've attempted to graph it, but I still could not find a minimum or maximum point within range. But aside from that, the problems assigned in the class are supposed to be done without calculators, and obviously without computer programs as well, so I have to figure out how to do it by hand. Thank you for the advice though.

 

[Ray Vickson]

I've attempted to graph it, but I still could not find a minimum or maximum point within range. But aside from that, the problems assigned in the class are supposed to be done without calculators, and obviously without computer programs as well, so I have to figure out how to do it by hand. Thank you for the advice though.

So, what background is available to you? Do you know trigonometric identities? If you do, you can re-write your function in the form $f(x) = A \sin(0.7 x + b)$ or as C \cos(0.7 x + r)## involving computable constants A,b or C,r. In those forms the function is easy to analyze.

Anyway, you have already found one of the max or min points (even though you think you have not, for some reason). There are others, however.

 

[HallsofIvy]

Every continuous function has both maximum and minimum on a closed and bounded interval. They occur in the interior where the derivative is 0, in the interior where the derivative does not exist, or at an endpoint.

"I can't find a value of x within the range."

Be sure your calculator is in "radian mode"! The derivative is zero at x= 1.965... and x= 6.452... which are in the given range. Find the value of the function at those values of x and at the endpoints.

 

[bah83]

Every continuous function has both maximum and minimum on a closed and bounded interval. They occur in the interior where the derivative is 0, in the interior where the derivative does not exist, or at an endpoint.

"I can't find a value of x within the range."

Be sure your calculator is in "radian mode"! The derivative is zero at x= 1.965... and x= 6.452... which are in the given range. Find the value of the function at those values of x and at the endpoints.

Thank you so much! You were absolutely correct. I was really confused by the way it was explained so I didn't think to turn on radian mode. Thank you.

 

[Ray Vickson]

Thank you so much! You were absolutely correct. I was really confused by the way it was explained so I didn't think to turn on radian mode. Thank you.

I am confused. Before you said "...the problems assigned in the class are supposed to be done without calculators...". So, is it OK to use a calculator, or not? (In my opinion such a no-calculator, no-computer restriction is silly in today's world, but you need to go by whatever restrictions are imposed on you.)

 

[vanhees71]

I must admit, I wouldn't know how to do this question without a calculator, but I haven't thought about the question much ;-).