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## AP®︎/College Calculus AB

### Course: AP®︎/College Calculus AB>Unit 5

Lesson 4: Using the first derivative test to find relative (local) extrema

# Worked example: finding relative extrema

AP.CALC:
FUN‑4 (EU)
,
FUN‑4.A (LO)
,
FUN‑4.A.2 (EK)
Sal finds the relative maximum point of g(x)=x⁴-x⁵ by analyzing the intervals where its derivative, g', is negative or positive.

## Want to join the conversation?

• So, Sal keeps mentioning that if the derivative is undefined, then the point is a critical point. Does that mean that if the slope of the tangent line is vertical at that given point, does that make that point a critical point?
• If the tangent line to a point (x, y) is vertical, then the derivative (slope of the vertical line) is undefined, so the function has a critical point at (x, y).

A critical point occurs wherever the derivative is either 0 or undefined. All extrema of a function will occur at critical points, though not all critical points will end up being extrema.
• Ok, so if the the derivative of f(x) is a fraction, how woud I find the critical numbers? Do i set the numerator and denominator equal to 0?
ex) the derivative of f(x) = (-4x+4)/(x+4)^4
• A critical point is a point where the derivative equals 0 or undefined so you would set the numerator equal to zero to find where f'(x)=0 and the denominator equal to 0 to find f'(x)=undefined
• How would you determine if an endpoint on a closed interval is a local minimum or local maximum?
• At , what exactly is the significance of Sal's "critical points"?
• You have to use the "critical points" to find the relative extrema.
(1 vote)
• Why 4/5 is a relative max? If the function increases from (0:4/5)? So 4/5 MUST BE GLOBAL MAX
• Global maximums are also relative maximums, so 4/5 is both a relative maximum and a global maximum. This is because the requirement for relative maximums is met where there is a global maximum (f' goes from f'>0 to f'<0).
(1 vote)
• What about if the two critical numbers found are the same value? How would one go about finding the relative extrema?
(1 vote)
• If two critical points occur at the same x value then it is actually just one and the same critical point. So you only have to check that one point.
(1 vote)
• How do you know you've covered all the possibilities for the critical points?
(1 vote)
• If you have an understanding of where the asymptotes are as well where the gradient is undefined or equal zero. So it helps to have an understanding of curve is its combination of two functions, is some conic section, some algebraic expression.

It basically through having an understanding of differentiation and limits that you ensure all critical points as at times it could be a bit difficult.
(1 vote)
• if we take double derivative and then put critical point in. its either concave up or down
if concave up then its relative min at that point, if concave down then relative max.
at zero i can't use double derivative.
what am i curious about is that "Is it work for all functions or for some function"
(1 vote)
• It bothers me how he keeps giving us a graph with a drawing that has one of Xs having f'(x) = undefined, yet doesn't give us an equation that could result to the drawing of a graph that has a "pointy" extrema.....