Just as the first derivative can be used to find the
extrema (maximum and minumum) values of a function near a
critical point, the second derivative can be used to find the
shape (concavity) of a function. The second derivative is the
rate of change of the slope, and if the second derivative changes
sign from negative to positive, the function's slope is
increasing upward. In that case, the function is said to be
concave upward on that interval. If the second derivative changes
sign and is negative, the function is said to be concave downward
on that interval.

Here we use the second derivative to determine the concavity of the function: $f\left(x\right)={x}^{3}-9{x}^{2}-12x+4$

Here we use the second derivative to determine the concavity of the function: $f\left(x\right)={x}^{3}-9{x}^{2}-12x+4$

Second derivative and tests for concavity

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