Math Is Fun Forum

technique09

Member

Registered: 2008-10-04

Posts: 14

Second Derivative

Hi there, i was wondering if someone could help me understand the second derivative in more detail.

Well, i understand that the second derivative means the rate of change of the rate of change, however what does it mean once you use it to solve problems? Here is my example:

"Find the stationary points of f(x)= x^4 - (6x)^2 + 8x and hence classify as local maxima, local minima or point of inflection.

f'(x)= (4x)^3 - 12x + 8      and therefore the stationary points are x=1 and x=-2.
f''(x)= (12x)^2 - 12           and thus f''(-2)=36 (which is positive which means x=-2 is a local minima.

My question is what is the significance of the 36? What does its value represent?

I apologise if anything ive written is unclear. I would like to thank anyone who has any helpful responses.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Second Derivative

First, remember that you're looking at the second derivative at a point where the first derivative is 0.

If the second derivative is positive, that means that the first derivative is increasing.
This means that it was negative when x was just below the critical point, and positive when x goes above the critical point.

This in turn means that the curve was going downwards before it became stationary, and then went upwards. Therefore, the critical point is a minimum.

Hopefully that made sense. You can use similar reasoning to explain why f'(x)=0 and f''(x) negative mean that x is a maximum.

Also, just to let you know, you shouldn't have any brackets around the things being taken to powers.

For example, (6x)² would be 6²*x² = 36x², which isn't what you want.

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technique09

Member

Registered: 2008-10-04

Posts: 14

Re: Second Derivative

That did make a lot of sense, i was thinking about the second derivative in exactly the same way as you are. However, in the question i was asking what the significance of the 36 was, what does the actual number represent? I understand how to get that number, but was is it showing? I hope im making sense in the questions im asking, its just been bugging me thinking about it.

PS Thanks for noting the mistakes with my brackets, it was careless.

luca-deltodesco

Member

Registered: 2006-05-05

Posts: 1,470

Re: Second Derivative

just as the first derivitive is the rate of change of the function, the second derivitive is the rate of change of the first. so that number 36 is how quickly the gradient is changing.

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technique09

Member

Registered: 2008-10-04

Posts: 14

Re: Second Derivative

Ahh ok, thanks made that much clearer for me. So basically, the number 36 in the question means nothing, however if a similar sort of question was applied to a real life problem, then it would have a meaning?

luca-deltodesco

Member

Registered: 2006-05-05

Posts: 1,470

Re: Second Derivative

in physics, acceleration is the second derivitive with respect to time of displacement, so in that case if you had a function for displacement in terms of time, the second derivitive is the acceleration which is useful.

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The Beginning Of All Things To End.
The End Of All Things To Come.

technique09

Member

Registered: 2008-10-04

Posts: 14

Re: Second Derivative

yeah i understand what you mean, thanks luca-deltodeco and mathsyperson for helping clear that up for me.