Increasing, decreasing and classifying turning points.
Example 1
f(x)=x2−4x+5. Find the x−coordinate of its stationary point.
Show worked answer
Answer: 2
- f′(x)=2x−4.
- Set f′(x)=0:x=2.
Example 2
f′(x)=−(x+2). On which interval is f decreasing?
f′(x)
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f′(x)=−(x+2)Show worked answer
Answer: x>−2
- A function is decreasing where its derivative is negative.
- Solving −(x+2)<0 gives x>−2; dividing by a negative coefficient reverses the inequality.
Example 3
For f(x)=−5(x+3)2−2, classify the stationary point at x=−3.
y
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f(x)=−5(x+3)2−2Show worked answer
Answer: Local maximum
- f′(x)=−10(x+3), so f′(−3)=0.
- The derivative changes from positive to negative at x=−3.
- The function rises then falls, so this is a local maximum.