- To find the second derivative, simply take the derivative of the original derivative.
- Given y=f(x), f ‘(x) represents the derivative function and represents the second
derivative.
- If f ’(x)<0 on an interval, then f(x) is decreasing on that interval.
- If f ’(x)>0 on an interval, then f(x) is increasing on that interval.
- If f ’ ’(x)<0 on an interval, then f ’(x) is decreasing on that interval and f(x) is concave down.
- If f ’ ’(x)>0 on an interval, then f ’(x) is increasing on that interval and f(x) is concave up.
Maxima and Minima
In order to continue with learning applications of derivatives, we must know key terms that will help us evaluate graphs and problems:
Definitions
critical point- for any function f, a point p in the domain of f where f ’(p)=0 or f ’(p) is undefined is called a critical point
critical value- a critical value of f is the value f(p), or the output, at a critical point p.
inflection point- a point at which the graph of a function changes concavity. At the inflection point, f ‘ ’(p) is either zero of undefined, and the function’s second derivative changes signs.
local maximum- a function f has a local maximum at p if f(p) is greater than or equal to the values of f for points near p.
local minimum- a function f has a local minimum at p if f(p) is less than or equal to the values of f for points near p.
global maximum- the single greatest value of a function f over a specified domain. A function f has a global maximum at p if f(p) is greater than or equal to all values of f.
global minimum- the single least value of a function f over a specified domain. A function f has a global minimum at p if f(p) is greater than or equal to all values of f.
local extrema or critical points- if a function has a local maximum or minimum that is not an endpoint on an interval, then the derivative of that point equals zero, and that point is a critical point. Remember, every local max or min is a critical point but not every critical point is a local max or min.
To find the global max or min of a continuous function on a closed interval, compare the values of the function at all critical points in the interval and at the endpoints.
Solving maxima/minima problems:
1. Find the derivative of the original function.
2. The points where x=0 are critical points, so either factor and find x or use the quadratic formula.
3. Plug in numbers in around the critical points to find out whether or not the points are local maxima or minima.
4. Take the second derivative and find where x=0, that is the inflection point(s).
5. Plug critical points and endpoints (if any) into the original function to find any global maxima or minima.
In order to continue with learning applications of derivatives, we must know key terms that will help us evaluate graphs and problems:
Definitions
critical point- for any function f, a point p in the domain of f where f ’(p)=0 or f ’(p) is undefined is called a critical point
critical value- a critical value of f is the value f(p), or the output, at a critical point p.
inflection point- a point at which the graph of a function changes concavity. At the inflection point, f ‘ ’(p) is either zero of undefined, and the function’s second derivative changes signs.
local maximum- a function f has a local maximum at p if f(p) is greater than or equal to the values of f for points near p.
local minimum- a function f has a local minimum at p if f(p) is less than or equal to the values of f for points near p.
global maximum- the single greatest value of a function f over a specified domain. A function f has a global maximum at p if f(p) is greater than or equal to all values of f.
global minimum- the single least value of a function f over a specified domain. A function f has a global minimum at p if f(p) is greater than or equal to all values of f.
local extrema or critical points- if a function has a local maximum or minimum that is not an endpoint on an interval, then the derivative of that point equals zero, and that point is a critical point. Remember, every local max or min is a critical point but not every critical point is a local max or min.
To find the global max or min of a continuous function on a closed interval, compare the values of the function at all critical points in the interval and at the endpoints.
- First derivative test for local maxima and minima:
- If f ’(x) changes from positive to negative at a critical point, then that critical point is a
local maximum.
- If f ’(x) changes from negative to positive at a critical point, then that critical point is a local minimum.
- If f ’(x) does not change signs at a critical point, then the critical point is neither a local maximum or a local minimum.
- Second derivative test for local maxima and minima:
- • If f ’ ’(x) is positive at a critical point, then that critical point is local minimum.
• If f ’ ’(x) is negative at a critical point, then that critical point is a local maximum. • If f ’ ’(x)=0 at a critical point, then the test tells us nothing.
Solving maxima/minima problems:
1. Find the derivative of the original function.
2. The points where x=0 are critical points, so either factor and find x or use the quadratic formula.
3. Plug in numbers in around the critical points to find out whether or not the points are local maxima or minima.
4. Take the second derivative and find where x=0, that is the inflection point(s).
5. Plug critical points and endpoints (if any) into the original function to find any global maxima or minima.