Detectives Detectives
This is a point where f′(x)=0 or is undefined.
What is a critical point?
If f′(x) changes from positive to negative at ccc, you found this.
What is a local maximum?
If test values around x=4 show f′(x) goes +→−, the point is this.
What is a local max?
Process: find f′(x), set it = 0, make sign chart, classify points.
What is the First Derivative Test?
We check both sides of a point like we're doing this before crossing the street.
What is “look both ways first”?
If f′(x)>0 on an interval, the function is doing this.
What is increasing?
If f′(x) changes from negative to positive at ccc, you found this.
What is a local minimum?
If test values around x=-2 show f′(x) goes -→+, the point is this.
What is a local min?
for critical
x/(x^2 -6x +16)
points occur here
What is x=−4 and x=4
This place can beat even a critical point for “best or worst value.”
What are the endpoints?
For polynomial functions, the derivative is undefined this many times.
What is never? (they're defined everywhere)
When f′(x) doesn't change sign, the point is this.
What is neither a max nor a min?
The sign of f′(x) determines whether the function is doing this.
What is increasing or decreasing?
For f(x)=2x3-15x2+24x+7 on [0,6], the abs max occurs here
What is x=6?
The derivative shows how you're moving, kind of like this app.
What is Google Maps / GPS?
Solving 6x2-30x+24=0 gives you these special x-values.
What are x=1 and x=4?
For f(x)=2x3-15x2+24x+7, the local max happens here.
What is x=1?
A horizontal tangent occurs when this equals 0.
What is f′(x)?
Given Intervals: f'(x)>0 on (-∞,1), f'(x)<0 on (1,4), f'(x)>0 on (4,∞)
What are a local max at x=1 and a local min at x=4?
A point where everything pauses and decides what to do next.
What is a dramatic pause / plot twist moment?