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The Sign of the Second Derivative

The Second Derivative

We can decide whether a function is increasing or decreasing by looking at the sign of its derivative. To decide whether f ' is increasing or decreasing, we must look at the sign of its derivative (the derivative of f '). This derivative of f ' (i.e. the derivative of the derivative of f) is called the second derivative of f and is denoted by f ''.

Example

If

f(x) = 3x3 + 11x2 + x + 1

then

f '(x) = 9x2 + 22x + 1

and

f ''(x) = 18x + 22.

Exercise

Calculate the second derivative of:

f(x) = x4 + x3 + x2 + x +

 

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