7C - Stationary points
14G - Content video: The nature of stationary points [From Unit 2]
This video covers the theory and an example of how to determine the nature of stationary points for a graph.
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A stationary point occurs when the gradient of a tangent line to a curve is zero.
- To find the \(x\)-ordinate of a stationary point on the curve of \(f(x)\), solve \(f'(x)=0\) for \(x\).
- To find the \(y\)-ordinate of a stationary point, subsitute the \(x\)-value into the original function, \(f(x)\).
Types of stationary points
There are three types of stationary points:
Determining the nature of a stationary point
They nature of a stationary point can be determined using any of the three methods below:
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7C - Example 1: Determining the nature of a stationary point (VCAA, 2010)
[VCAA, 2010 Exam 2 MC Question 17]
The function \(f\) is differentiable for all \(x\in R\) and satisifes the following conditions.
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7C - Example 1: Video solution
7C - Example 1: Practice
Question 1 [IB, 2008]:
If \(f(x)=x-3x^{\frac{2}{3}}\), where \(x>0\).
7C - Example 1: Solutions
Question 1: (a) \(x=8\) (b) \(P\) is a local minimum stationary point. |