Differentiation: Applications of derivatives
Inflection points
An inflection point, or point of inflection, is a point where the graph of a function makes a kind of "bend" by changing the type of increase or decrease. This means that the derivative of the function changes from increasing to decreasing, or vice versa, and thus the derivative has a local maximum or minimum. These points can only occur if the second derivative is #0#.
An inflection point of a graph is a point in which the concavity of the function changes.
In other words, it is a point where the graph changes from concave up to concave down, or from concave down to concave up.
We can find the inflection points using the second derivative.
A function #\blue{f(x)}# has an inflection point at #x=\orange{c}# if #\purple{f''(}\orange{c}\purple{)}=0# and #\green{f'(}\orange{c}\green{)}# is a local maximum or minimum of #\green{f'(x)}#.
Example
Step-by-step Calculating inflection points |
Example |
|
Determine the inflection points for a function #f(x)#. |
#\begin{array}{rcl}f(x)&=&\phantom{'}x^5-x^3\end{array}# |
|
Step 1 |
Calculate the first derivative #f'(x)#. |
#\begin{array}{rcl}f'(x)&=&5x^4-3x^2\end{array}# |
Step 2 |
Calculate the second derivative #f''(x)#. |
#\begin{array}{rcl}f''(x)&=&20x^3-6x\end{array}# |
Step 3 |
Solve #f''(x)=0# to find the possible local minima and maxima of #f'(x)# and with them the possible inflection points of #f(x)#. |
#\begin{array}{c}{x}={0} \lor {x}={\sqrt{\tfrac{3}{10}} }\;\lor\; {x}={-\sqrt{\tfrac{3}{10}}}\end{array}# |
Step 4 |
Determine whether the values found in step 3 belong to a local minimum or maximum of #f'(x)#. If so, then they are inflection points. |
#{x}={-\sqrt{\tfrac{3}{10}}}# local minimum #f'# #{x}={0}# local maximum #f'# #{x}={\sqrt{\tfrac{3}{10}}}# local minimum #f'# Hence all three are inflection points #f(x)# |
Give your answer in the form #x=x_1 \lor x=x_2 \lor \ldots\lor x=x_n# if there are #n# number of inflection points, in the form #x=x_1# if there is one point of inflection, and in the form #none# if there are no inflection points.
Step 1 | We determine the derivative of #f(x)=x^3# using the power rule. This is equal to: \[f'(x)=3\cdot x^2\] |
Step 2 | We determine the second derivative in the same way. This is equal to: \[f''(x)=6\cdot x\] |
Step 3 | We solve the equation #6\cdot x=0#. This goes as follows: \[\begin{array}{rcl} 6\cdot x&=&0\\&&\phantom{xxx}\blue{\text{the equation we need to solve}}\\ x&=&0\\&&\phantom{xxx}\blue{\text{both sides divided by }6}\\ \end{array} \] |
Step 4 | We draw the graph of #f'(x)=3\cdot x^2#. We see that #f'(x)# has a local minimum at #x=0#. Therefore, the function #f(x)# has an inflection point at #x=0#. |
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