Tests and Theorems
If a function is continuous on a closed interval [a,b], then it must have both an absolute maximum and an absolute minimum somewhere on that interval.
Extreme Value Theorem, EVT
A point that is the lowest within a small neighborhood of the graph.
Local Minimum / Relative Minimum
Where the derivative is positive, the function is doing this.
f(x) is increasing
If f′′(x)>0, the graph is shaped like this.
Concave up
f′′(x)=6x−12,
find the x-value where an inflection point could occur.
x = 2
Check critical points + endpoints to find absolute max/min on a closed interval.
Candidates Test
A point that is the highest within a small neighborhood of the graph.
Local Maximum / Relative Maximum
Where the derivative is negative, the function is doing this.
f(x) is decreasing
If f′′(x)<0, the graph is shaped like this.
Concave Down
Let f(x) = x^3 -6x^2 + 9x + 1
Find the critical points
x = 1 and x = 3
Used to classify critical points by checking concavity.
If f′′(c)>0, you have a local minimum.
If f′′(c)<0, you have a local maximum.
Second Derivative Test
The highest value of the function on the entire interval.
Absolute Maximum / Global Maximum
A point where f′(x)=0 or undefined.
Critical Point
A point where concavity changes.
Inflection Point
f(x) = 5x^3-10x^2-5
is the function increasing or decreasing at x=2?
The function is increasing at x = 2
Used to determine local maxima or minima by looking at how the sign of f′(x) changes around a critical point.
First Derivative Test
The lowest value of the function on the entire interval.
Absolute Minimum / Global Minimum
If f′(x) changes from positive to negative, you have this.
local maximum
This derivative tells you concavity.
2nd Derivative f''(x)
f(x) = 5x^3 -10x^2+10x+5
is f(x) concave up or concave down at x = 5?
concave up
The __________________ in calculus states that for a function continuous on [a, b] and differentiable on (a, b), there’s a point c in (a, b) where the instantaneous rate of change (derivative, f′(c)) equals the average rate of change over the interval, f'(c) = (f(b) - f(a)) / (b-a)
MVT, Mean Value Theorem
A point where the derivative is zero or undefined but the function is not a max or a min.
Neither Maximum nor Minimum (Saddle Point / Critical Point with No Extremum)
If f′(x) changes from negative to positive, you have this.
local minimum
A function can only have an inflection point where this derivative is zero or undefined.
f''(x) = 0 or DNE
The sum of two positive numbers is 56.
Find the pair of numbers that gives the maximum product.
Two numbers: 28,28
Maximum product: 784