Finding Extreme Values on an Interval Using Analytic Methods | Calculus Help

[zaboda42]

Hey guys, i have the solution manual for my calc book but this problem set is really stumping me. I look at the examples, i look at the solution, i look at the steps but can't seem to see exactly what they're doing. Can anyone help?!

Use analytic methods to find the extreme values of the function on the interval and where they occur.

f(x) = sin (x + (pi/4)) , 0 <= x <= (7pi/4)

I know it's probably an easy problem and i don't really need to know the solution. I'm just having a really difficult time figuring out exactly what to do for problems like these. ANYONE care to explain?

Thanks!

 

[VeeEight]

Do you know any methods to find extrema of a function (perhaps involving a derivative)?

 

[zaboda42]

Yes,

I know you need to get the first derivative of the function and that's about it.

In the book they describe critical points and such and i have no idea what they mean

 

[Mark44]

Yes,

I know you need to get the first derivative of the function and that's about it.

In the book they describe critical points and such and i have no idea what they mean

In mathematics, definitions are crucial. How does your book define the term "critical point?"

 

[zaboda42]

A point in the interior of the domain of a function f at which f' = 0 or f' does not exist is a critical point of f.

I honestly just need to know how to do this problem. Please if anyone can help.

 

[Mark44]

For this function, there are no numbers for which f' doesn't exist, so where is f'(x) = 0 in the given interval? Those are your critical points.

 

[njama]

Correct me if I'm wrong but you need to find the values of x for which the function have min, max and =0 ?

Start with the fact that

-1 \leq sin(x) \leq 1

max=1, min=-1, zero=0

It same for sin(x + п/4).

sin(x+п/4)=1 (max)

sin(x+п/4)=-1 (min)

sin(x+п/4)=0 (zero)

Now, find the values for x.

 

[HallsofIvy]

A point in the interior of the domain of a function f at which f' = 0 or f' does not exist is a critical point of f.

I honestly just need to know how to do this problem. Please if anyone can help.

You have twice now talked about " f' ". Does it occur to you that you should find the derivative of sin(x+ \pi/4)? What is that derivative?

As to njama's remark, it is true that the relative max and min of a function must occur at critical points (not necessarily where the function value is 0) but the converse is not true. It is possible (though it doesn't happen for sine) that a function has a critical point where there is no max or min. For this function, determining where the function is 1 or -1 will give all critical points but that is not a general method.