C2 - Explaining the second derivative? Watch

TheKevinFang
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Looking at the explanation (attached), it says that the second derivative is the "rate of change of the gradient".

What does that mean? Could someone explain how it applies to the diagram below the explanation? On the lowest diagram, it gives values of d^2y/dx^2 on the y-axis; I can visualise that - if the value is 6x-12, then substituting the x-values, you get:

6(3) - 12 = 6 = minimum
6(1) - 12 = -6 = maximum

But why does the value on the y-axis mean the "rate of change of gradient"? Shouldn't it be the y-coordinate of the gradient at a given x-coordinate?

An explanation would be GREATLY appreciated, as I have an exam next week.

Kind regards,

Kevin.
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TheKevinFang
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Also, what happens if the value of d^2/dx^2 are both negative? e.g. one value is -6, and one value is -12.

Following logic, the value closer to 0 is the maximum point (-6), while the value which is lower (-12) is the minimum point?
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razzor
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(Original post by TheKevinFang)
Also, what happens if the value of d^2/dx^2 are both negative? e.g. one value is -6, and one value is -12.

Following logic, the value closer to 0 is the maximum point (-6), while the value which is lower (-12) is the minimum point?
The second derivative determines the nature of the stationary point (whether it is a minimum or maximum)

If it is negative, then the rate of change of the gradient is negative making the gradient decrease causing the stationary point to be a maximum.

Similarly, if it is positive at the stationary point, it is a minimum (the gradient of the curve is increasing from the stationary point)
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