Differentiation: Applications of derivatives
The second derivative
The derivative #f'# of a function #f# can be derived again. We call this the second derivative of #f#.
For a function #\blue{f(x)}# , we denote the second derivative as:
\[\green{f''(x)}=\frac{\dd}{\dd x}\orange{f'(x)}=\frac{\dd}{\dd x}\left(\frac{\dd}{\dd x}\blue{f(x)}\right)\]
Example
\[\begin{array}{rcl}\blue{f(x)}&\blue{=}&\blue{3x^2}\\ \orange{f'(x)}&\orange{=}&\orange{6x}\\\green{f''(x)}&\green{=}&\green{6}\end{array}\]
The second derivative is useful when one wants to find the extreme values of a function #f(x)#. We saw earlier that the condition #f'(c)=0# did not immediately imply that #c# corresponds to an extreme value. The following theorem will help us determining whether such a value corresponds to an extreme value or not.
If for a function #\blue{f(x)}# and a point #x=\purple{c}# we have
- #\orange{f'(}\purple{c}\orange{)}=0#
- #\green{f''(}\purple{c}\green{)}\neq 0#,
then, #\blue{f(x)}# has an extreme value in #\purple{c}#.
If #\green{f''(}\purple{c}\green{)}>0#, then #\purple{c}# corresponds to a local minimum. If #\green{f''(}\purple{c}\green{)}<0#, then #\purple{c}# corresponds to a local maximum.
Example
\[\begin{array}{rcl}\blue{f(x)}&=&\blue{2x^2+x}\\
\orange{f'(x)}&=&\orange{4x+1}\\
\green{f''(x)}&=&\green{4}\\
\orange{f'(}\purple{-\frac{1}{4}}\orange{)}&=&0\\
\green{f''(}\purple{-\frac{1}{4}}\green{)}&=&4\neq 0\end{array}\]
Simplify your answer as much as possible.
We first calculate the first derivative using the power rule.
\[f'(x)=6\cdot x^2-8\]
Then we calculate the second derivative in the same way.
\[f''(x)=12\cdot x\]
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