Differentiation: Applications of derivatives
Concavity
We previously saw that the first derivative of a function tells us when the function is increasing, stationary, or decreasing. However, if we think of the functions #f(x)=x# and #g(x)=x^2# for #x>0#, we see that both increase in different ways, with #y=x# doing so linearly and #y=x^2# curving upwards. The distinction between these behaviors can be made using the second derivative and a property called concavity.
A function #\blue{f(x)}# is said to be concave up at a point #x=\orange{c}# if #\purple{f''}(\orange{c})>0#.
Conversely, #\blue{f(x)}# is said to be concave down at a point #x=\orange{c}# if #\purple{f''}(\orange{c})<0#.
Notice that upward and downward concavity can happen independently of whether a function is increasing, decreasing, or stationary.
At a point #x=\orange{c}#, a function is
- concave up and increasing if #\purple{f''}(\orange{c})>0# and #\green{f'}(\orange{c})>0#,
- concave up and stationary if #\purple{f''}(\orange{c})>0# and #\green{f'}(\orange{c})=0#,
- concave up and decreasing if #\purple{f''}(\orange{c})>0# and #\green{f'}(\orange{c})<0#.
Similarly, at a point #x=\orange{c}#, a function is
- concave down and increasing if #\purple{f''}(\orange{c})<0# and #\green{f'}(\orange{c})>0#,
- concave down and stationary if #\purple{f''}(\orange{c})<0# and #\green{f'}(\orange{c})=0#,
- concave down and decreasing if #\purple{f''}(\orange{c})<0# and #\green{f'}(\orange{c})<0#.
Finally, we can define what it means for a function to be either concave or convex.
Concave and convex functions
A function #\blue{f(x)}# is said to be convex if #\purple{f''(x)}\geq 0# for all #x# in its domain.
A function #\blue{f(x)}# is said to be concave if #\purple{f''(x)}\leq 0# for all #x# in its domain.
Examples
#f(x)=e^x# is a convex function.
#f(x)=\ln(x)# is a concave function.
#f(x)=\cos(x)# is neither concave nor convex.
Here we see the function \[f(x)=\sin\left(\frac{1}{2}\pi\cdot x\right)+1\] This function is
- #\green{\text{concave down and increasing}}# between #0# and #1# (dashed part)
- #\orange{\text{concave down and decreasing}}# between #1# and #2# (dotted part)
- #\purple{\text{concave up and decreasing}}# between #2# and #3# (solid part)
- #\blue{\text{concave up and increasing}}# between #3# and #4# (long-short dashed part)
First we determine the first derivative using the power rule. That gives:
\[f'(x)=-6\cdot x^2+4\cdot x-6\]
We now substitute #x=-2# in #f'#. That gives:
\[f'(-2)=-6\cdot -2^2+4\cdot -2-6=-38\]
Next, we determine the second derivative by differentiating #f'# again using the power rule. That gives:
\[f''(x)=4-12\cdot x\]
We now substitute #x=-2# in #f''#. That gives:
\[f'' (-2)=4-12\cdot -2=28\]
Because #f'(-2) \lt 0# and #f''(-2) \gt 0# applies, #f(x)# is concave up and decreasing in the point #x=-2#.
Or visit omptest.org if jou are taking an OMPT exam.