Integral Calculus Question ? (please help me)

[hvmidwvli]

Find a function y = ƒ(x) having atleast three inflection points and sketch it's graph. [ Note: To find the formula of ƒ(x) start with a suitably chosen ƒⁿ(x) and integrate ƒⁿ(x) twice. The second part of the question is a curve sketching problem. Give all details. ]

 

[Harry_the_cat]

When do inflection points occur?

 

[Steven G]

You ask for help but it really seems that you are asking for someone to work out this problem for you. These two are not the same as someone solving this problem for you will not be very helpful for you.

I suggest that you read the forum's posting guidelines and go from there.

 

[hvmidwvli]

You ask for help but it really seems that you are asking for someone to work out this problem for you. These two are not the same as someone solving this problem for you will not be very helpful for you.

I suggest that you read the forum's posting guidelines and go from there.

The nature of the question is like that because I didn't change any of it's text. It is one of my homework questions that I couldn't solve. I cracked the question to a certain point according to my working but can't go furthur. I wanted someone to help me get to my answer. Read the website name.

I have read the Terms and Rules and it does not mention not being allowed to post homework questions as far as I can remember, so what is your point exactly I don't get it.

 

[Harry_the_cat]

The nature of the question is like that because I didn't change any of it's text. It is one of my homework questions that I couldn't solve. I cracked the question to a certain point according to my working but can't go furthur. I wanted someone to help me get to my answer. Read the website name.

I have read the Terms and Rules and it does not mention not being allowed to post homework questions as far as I can remember, so what is your point exactly I don't get it.

Jomo's point is that one of the terms and conditions is to:
Show your beginning work, or ask a specific question about the exercise, or explain why you're stuck
not just post the question and expect someone to give you the solution.

You also instruct (rather rudely IMHO) Jomo to "read the website name". The website name is "freemathhelp" not "freemathsolutions". We give great help here, but you have to put in a bit of effort too. Show us what you've done and where you get stuck so we have an idea where to begin helping you.

 

[hvmidwvli]

Jomo's point is that one of the terms and conditions is to:
Show your beginning work, or ask a specific question about the exercise, or explain why you're stuck
not just post the question and expect someone to give you the solution.

You also instruct (rather rudely IMHO) Jomo to "read the website name". The website name is "freemathhelp" not "freemathsolutions". We give great help here, but you have to put in a bit of effort too. Show us what you've done and where you get stuck so we have an idea where to begin helping you.

Okay if i show you guys my working will you consider helping me

 

[hvmidwvli]

Here's my working

 

[Dr.Peterson]

Here's my working

Your example has only one inflection point; that's because its second derivative has only one zero.

What you'd like is for the second derivative to have three zeros.

So what if f''(x) looked like what you used for f(x)? Can you find such a function?

That's what the hint tells you: integrate.

 

[hvmidwvli]

Your example has only one inflection point; that's because its second derivative has only one zero.

What you'd like is for the second derivative to have three zeros.

So what if f''(x) looked like what you used for f(x)? Can you find such a function?

That's what the hint tells you: integrate.

So the second derivative is supposed to have 3 inflection points, right? Or in other words 3 zeros.

So in that case I should construct a formula with a largest power of 5, right?

(Sorry for asking too many questions)

 

[Dr.Peterson]

So the second derivative is supposed to have 3 inflection points, right? Or in other words 3 zeros.

You didn't say what you mean, but I think you meant that the function f has to have 3 inflection points, so the second derivative has to have 3 zeros.

So in that case I should construct a formula with a largest power of 5, right?

Yes. And they told you how to construct it: integrate the function you chose as second derivative. Do you understand why?

 

[Steven G]

Okay if i show you guys my working will you consider helping me

Absolutely, just like the name of the website suggests.

 

[Steven G]

It is not as easy as just having the 2nd derivative equaling zero.

Consider the function f(x) = x^3. Then f"(x) = 6x which equals 0 when x=0. There is a point of inflection at x=0.

Now consider the function f(x) =x^4. Then f"(x) = 12x^2 which equals 0 when x=0. However, there is no point of inflection at x=0.

Draw those two functions to see for yourself!!

Conclusion, just because f"(x)=0 for some x-values does not guarantee that there is a POI at that x-value.

Now the obvious question to ask is, if f"(x) = 0 for some x-value, then how do we tell that there is a POI at this x-value.

The answer is simple. If f"(x=c) = 0 and f(c) has a sign change around x=c, then f(x) has a point of inflection at x=c. Otherwise it does not have a POI at x=c.

In the example above where f(x) = x^3 we did have a sign change at x=0 in the 2nd derivative. f"(x) = 6x. If x<0, f"(x) < 0 and if x>0, then f"(x)>0. There is a sign change for the 2nd derivative at x=0, so there is a POI at x=0.

This is different for f(x)=x^4. Yes, f"(0) =0 but there's no POI at x=0. Why?? f"(x) = 12x^2. If x<0 or x>0, f"(x)>0. That is there was no sign change at x=0 for the 2nd derivative, meaning that there is no POI at x=0 (even though f"(x)=0 when x=0).

 

[Steven G]

So the second derivative is supposed to have 3 inflection points, right? Or in other words 3 zeros.

So in that case I should construct a formula with a largest power of 5, right?

(Sorry for asking too many questions)

No, the polynomial function can have a power larger than 5 and still have only three points of inflections.
Also, functions other than polynomials have points of inflections! Consider y=sin x, it has an infinite number of POI. You can just restrict the function to an interval that has exactly three points of infections and be done.

 

[blamocur]

It is not as easy as just having the 2nd derivative equaling zero.

Consider the function f(x) = x^3. Then f"(x) = 6x which equals 0 when x=0. There is a point of inflection at x=0.

Now consider the function f(x) =x^4. Then f"(x) = 12x^2 which equals 0 when x=0. However, there is no point of inflection at x=0.

Draw those two functions to see for yourself!!

Conclusion, just because f"(x)=0 for some x-values does not guarantee that there is a POI at that x-value.

Now the obvious question to ask is, if f"(x) = 0 for some x-value, then how do we tell that there is a POI at this x-value.

The answer is simple. If f"(x=c) = 0 and f(c) has a sign change around x=c, then f(x) has a point of inflection at x=c. Otherwise it does not have a POI at x=c.

In the example above where f(x) = x^3 we did have a sign change at x=0 in the 2nd derivative. f"(x) = 6x. If x<0, f"(x) < 0 and if x>0, then f"(x)>0. There is a sign change for the 2nd derivative at x=0, so there is a POI at x=0.

This is different for f(x)=x^4. Yes, f"(0) =0 but there's no POI at x=0. Why?? f"(x) = 12x^2. If x<0 or x>0, f"(x)>0. That is there was no sign change at x=0 for the 2nd derivative, meaning that there is no POI at x=0 (even though f"(x)=0 when x=0).

If f"(x=c) = 0 and f(c) has a sign change around x=c, then f(x) has a point of inflection at x=c. Otherwise it does not have a POI at x=c.

I am guessing you mean "... and f'(c) has a sign change around x=c...", i.e., the first derivative must changes the sign, not the function itself.

 

[Steven G]

I am guessing you mean "... and f'(c) has a sign change around x=c...", i.e., the first derivative must changes the sign, not the function itself.

Thanks for pointing out my error for the OP.
However we are both wrong!
It should be---If f"(x=c) = 0 and f''(c) has a sign change around x=c, then f(x) has a point of inflection at x=c. Otherwise it does not have a POI at x=c.

 

[blamocur]

We are both wrong!
It should be---If f"(x=c) = 0 and f''(c) has a sign change around x=c, then f(x) has a point of inflection at x=c. Otherwise it does not have a POI at x=c.

Ooops