C2 Maximum value of f(x)

Greetings,

I'm revising through solomon paper and I have the following problem:

f(x)=4/(2+sinx)

(i) State the maximum value of f(x) and the smallest positive value of x for which f(x) takes this value.

I did some research but I have no idea how to solve it. Can anyone guide me through this please? Thanks indeed.

Think about the values sin x can take and which values of x produce these critical values.

Reply 2

dragonkeeper999

17

Differentiate f(x). At turning points, f'(x) = 0. At maximum points, f''(x) < 0. Sine waves are repetitive, so you want the maximum point which has the lowest x value greater than 0.
Edit: yeah, Mr M's method is easier!

Reply 3

james22

16

You have a fraction so to maximise it you want to minimize the denominator. Can you do that?

Reply 4

Blackfyre

OP

10

As I mentioned above I have no idea about any of this, that's why I need to be guided through the whole exercise so I can understand it fully. Thanks.

Original post by Blackfyre

As I mentioned above I have no idea about any of this, that's why I need to be guided through the whole exercise so I can understand it fully. Thanks.

That's not how we do it on TSR. You need to do some of the work.

I'll ask again. Do you know the values sin x can take?

Reply 6

LegendX

15

Original post by Blackfyre

Greetings,

I'm revising through solomon paper and I have the following problem:

f(x)=4/(2+sinx)

(i) State the maximum value of f(x) and the smallest positive value of x for which f(x) takes this value.

I did some research but I have no idea how to solve it. Can anyone guide me through this please? Thanks indeed.

Imagine a sine graph, what is the biggest value of that a sine graph can produce? Now think of that graph but translated up by a factor of 2.

Original post by dragonkeeper999

Edit: yeah, Mr M's method is easier!

Not just easier - this is C2 so Blackfyre probably hasn't met the quotient rule or trigonometric differentiation yet.

Original post by LegendX

translated up by a factor of 2.

Translations do not have factors.

Reply 9

dragonkeeper999

17

Original post by Mr M

Not just easier - this is C2 so Blackfyre probably hasn't met the quotient rule or trigonometric differentiation yet.

oops, didn't think about that!

Original post by LegendX

what is the biggest value of that a sine graph can produce

We actually want the smallest value.

Reply 11

Blackfyre

OP

10

Original post by Mr M

That's not how we do it on TSR. You need to do some of the work.

I'll ask again. Do you know the values sin x can take?

I'm up to do it by myself or study by myself so I can fully understand it but I can't seem to find anything related to the maximum value of f(x) in the book, that's why I need some guidance through this. I don't know the values sinx can take.

Do I understand it right, do I have to change the form of the function so it's not a fraction anymore and then differentiate/integrate it and equal to zero (like finding maximum/minimum point with differentiation)?

Original post by Blackfyre

I'm up to do it by myself or study by myself so I can fully understand it but I can't seem to find anything related to the maximum value of f(x) in the book, that's why I need some guidance through this. I don't know the values sinx can take.

Do I understand it right, do I have to change the form of the function so it's not a fraction anymore and then differentiate/integrate it and equal to zero (like finding maximum/minimum point with differentiation)?

You don't need to do any of that.

Draw the graph of y=sin x for

0 \leq x \leq 2\pi

(I am assuming you understand radian measure). Can you do that?

Edit: I've done it for you.

Now what is the lowest value sin x takes and what is the exact value of x (as a multiple of

\pi

) that produces this minimum?

(edited 11 years ago)

Reply 13

Slowbro93

20

Original post by Blackfyre

I'm up to do it by myself or study by myself so I can fully understand it but I can't seem to find anything related to the maximum value of f(x) in the book, that's why I need some guidance through this. I don't know the values sinx can take.

Do I understand it right, do I have to change the form of the function so it's not a fraction anymore and then differentiate/integrate it and equal to zero (like finding maximum/minimum point with differentiation)?

You're overcomplicating it. Forget the question and think of the sin curve. What is the range of the function (so the minimum to maximum value)?

Reply 14

Blackfyre

OP

10

Original post by Mr M

You don't need to do any of that.

Draw the graph of y=sin x for

0 \leq x \leq 2\pi

(I am assuming you understand radian measure). Can you do that?

Edit: I've done it for you.

Now what is the lowest value sin x takes and what is the exact value of x (as a multiple of

\pi

) that produces this minimum?

Thanks for drawing the graph even though I have no problems with it. I don't understand what the question is asking me to do. I do understand the radian measure but it's asked in degrees. Sorry if my stupidity is killing you but honestly I have trouble understanding what the question is asking me to do.

Original post by Blackfyre

Thanks for drawing the graph even though I have no problems with it. I don't understand what the question is asking me to do. I do understand the radian measure but it's asked in degrees. Sorry if my stupidity is killing you but honestly I have trouble understanding what the question is asking me to do.

Ok your question did not say anything about degrees and it is always assumed that you are using radians unless you are told otherwise.

So what is the lowest value of sin x and what is the value of x in degrees that produces this minimum?

Reply 16

Kerch

8

Original post by Blackfyre

Thanks for drawing the graph even though I have no problems with it. I don't understand what the question is asking me to do. I do understand the radian measure but it's asked in degrees. Sorry if my stupidity is killing you but honestly I have trouble understanding what the question is asking me to do.

There's a certain value of x which will make f(x) a maximum (actually there are infinitely many because sin(x) is periodic).

Because f(x) is in the form of a fraction, its maximum occurs when the denominator is minimised (as the denominator is positive in this case).

Your denominator is 2 + sin(x), so from knowledge of what sin(x) looks like you can work out the value(s) of x that make 2 + sin(x) a minimum.

Reply 17

Blackfyre

OP

10

Original post by Mr M

Ok your question did not say anything about degrees and it is always assumed that you are using radians unless you are told otherwise.

So what is the lowest value of sin x and what is the value of x in degrees that produces this minimum?

I don't understand what is meant by minimum and maximum value, as I said in the post above I don't understand what's the question asking me to do.

Original post by Blackfyre

I don't understand what is meant by minimum and maximum value, as I said in the post above I don't understand what's the question asking me to do.

Minimum = smallest

Maximum = biggest

The key to solving this is to look at this diagram and answer this question:

What is the lowest y value the sin x graph takes and at what x value does this occur?

Edit: I deliberately used the words lowest and smallest previously so that the language of the question would not cause difficulties.

(edited 11 years ago)

for the first bit, think of it like finding the maximum value of a "fraction"

if you have

\frac{2}{a}

the smaller the a, the larger the whole fraction - like:

\frac{1}{5}>\frac{1}{10}>\frac{1}{20}

so, to maximise the function, you have to ask yourself: what value does the Sine part take (or, what values is it allowed to take) to make the denominator as small as it can be ?

the Sine function itself does take negative values down to a minimum of -1 which will give the function a certain value:

set the f(x) equal to this value and solve in the range: [0,2Pi]

(edited 11 years ago)

C2 Maximum value of f(x)

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