Trig identities: given |x| = π / 4 What's the minimum value of: f(x) = (1 / (sec^2 x)) + sin x ?

hearts123

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Hi everyone :)
Here's a question I'm having some trouble with:

given |x| = π / 4
What's the minimum value of: f(x) = (1 / (sec^2 x)) + sin x ?

I used desmos to graph both of these and this is what I got: https://ibb.co/QY9bDJs (is this type of image link acceptable?)
Would the leftmost point within the red highlighted part be the minimum value?

How can I get the minimum value without using graphing, and instead by using identities and such?
Thanks so much in advance!
 

MarkFL

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It appears you mean:

\(\displaystyle |x|\le\frac{\pi}{4}\)

or:

\(\displaystyle -\frac{\pi}{4}\le x\le\frac{\pi}{4}\)

Have you studied differential calculus?
 

mmm4444bot

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https://ibb.co/QY9bDJs (is this type of image link acceptable?)
Yes, you may post an URL to hosted material, as long as the hosting site doesn't add anything that violates forum policy (eg: aggressive advertising, inappropriate additional images). The imgbb site is a good choice, and they also offer pre-formatted versions of code for use in forums -- some of which may display the image in your post.

😎
 

hearts123

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It appears you mean:

\(\displaystyle |x|\le\frac{\pi}{4}\)

or:

\(\displaystyle -\frac{\pi}{4}\le x\le\frac{\pi}{4}\)

Have you studied differential calculus?
No, I haven't studied that. Are the two inequality/ranges the same? In the question, I was given the first one (absolute value of x is lesser or equal to pi/4)
 

MarkFL

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Yes, the two inequalities I posted are equivalent, but the first (what you were given) is a more succinct statement. I think I would begin by restating the given objective function as:

\(\displaystyle f(x)=\cos^2(x)+\sin(x)\)

However, without the aid of the calculus, I would be hard pressed to find the local extrema. Were you instructed to simply use a computer generated graph to aid you?
 

Dr.Peterson

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One thought is to take MarkFL's rewrite and express it as a quadratic function of sin(x). You can find the maximum of that, and use the value of sin(x) that produces that maximum to solve for x. This doesn't require calculus, though it does require some additional thought to be sure you haven't missed anything.

But notice that, as you can see from your graph, the minimum is at an endpoint of the domain; so the work I suggested doesn't actually lead to the answer, just a better understanding of the graph!
 

Dr.Peterson

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given |x| π / 4
What's the minimum value of: f(x) = (1 / (sec^2 x)) + sin x ?

I used desmos to graph both of these and this is what I got: https://ibb.co/QY9bDJs
Would the leftmost point within the red highlighted part be the minimum value?

How can I get the minimum value without using graphing, and instead by using identities and such?
I just realized no one has actually answered your questions.

Yes, graphically you have found the minimum as f(-π / 4), the leftmost point in the domain. Evaluate that, and you have the answer.

Ultimately, you probably need to use graphing techniques (including calculus to find the shape, if you are working by hand) at least as an aid. I don't think identities and algebra alone will do the trick, though as I showed they can play a role.
 
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