Graphing Question (1 Viewer)

[weirdguy99]

Hi guys,

I almost got this question right from the 2009 NSGHS Trials.

9 (b): A continuous function is defined by the following features.

y'' > 0 for x < -1 and 1 < x < 3 .

y' = 0 only when x = -3, 1, and 5 .

y = 0 only when x = 1.

Sketch a possible graph of the function.

I took y'' > 0 as it being a minimum turning point at x = -3 and 5. At x = 0, my graph hit the x-axis. I basically drew x^4 sort of curve, with x = 1 being the minimum turning point. But the curve is actually a maximum at x = -3, a minimum at x = 5 and a point of inflexion (?) at y = 1.

So..how do I do it?

Thanks!

 

[SpiralFlex]

Hi guys,

I almost got this question right from the 2009 NSGHS Trials.

9 (b): A continuous function is defined by the following features.

y'' > 0 for x < -1 and 1 < x < 3 .

y' = 0 only when x = -3, 1, and 5 .

y = 0 only when x = 1.

Sketch a possible graph of the function.

I took y'' > 0 as it being a minimum turning point at x = -3 and 5. At x = 0, my graph hit the x-axis. I basically drew x^4 sort of curve, with x = 1 being the minimum turning point. But the curve is actually a maximum at x = -3, a minimum at x = 5 and a point of inflexion (?) at y = 1.

So..how do I do it?

Thanks!

I drew you some pretty diagrams.

Let's have a look. We are given only . We know that it will only cross the axis at . Now, obviously, at they will either be maximum or minimum, we do not know yet.

So now it says for and . So what does this exactly mean? It means that the concavity is positive, hence there is a smily face.

Focusing on second derivative (Concavity)

Remember it said (Positive concavity, facing up, smiley face)

Step 1: So when it will be a smile. So we know that it's going to have a smiley shape. Now that region is taken care of.

Step 2: So when we get another positive concavity. Hence another smiley shape. Your graph should now look like,





Left over regions

We wouldn't expect our graphs to have another positive concavity in the left over regions. We would expect a negative concavity.

Step 3: Now our left over regions of must have a negative concavity. (Sad face. ) [We can note an horizontal inflexion at . Change of concavity.]

Step 4: Another region left over is . This is to have a negative concavity too. (Sad face ) [We can note a inflexion at . Change of concavity.]

Finally our graph should look something like this drawing in our negative concavities. Make sure you indicate that this function is continuous with nice arrow heads.

 

[weirdguy99]

Well, you got the graph right, but I don't see how it's a minimum at x = 3.

 

[SpiralFlex]

Well, you got the graph right, but I don't see how it's a minimum at x = 3.

I am not fully done yet. Sorry, it's late.

Well, you got the graph right, but I don't see how it's a minimum at x = 3.

Which part do you not understand?

 

[weirdguy99]

So you just work out the areas which are prescribed, and then basically fill in the blanks?

 

[SpiralFlex]

So you just work out the areas which are prescribed, and then basically fill in the blanks?

Yes. You need to reason why certain parts are like that. For example How did I know it was a negative concavity? Since we are given the region is positive and only stationary points are We know that must be a maximum since it must be a stationary point. Think about if it was minimum, it would be a positive concavity, the only positive concavities mentions are between the regions stated. Certainly is not a point in that region.

Sorry if I have confused you. Do you need me to clarify anything or can I go play outside now?

 

[weirdguy99]

I GET IT! THANKSSSSS!!

Also, you may go outside now

 

[SpiralFlex]

I GET IT! THANKSSSSS!!

Also, you may go outside now

Thank you human! SpiralFlex is off hunting some delicious humans while they sleep. Hopefully I can make it back before sunrise to eat you, I mean help you...