Increasing/ Decreasing function Flashcards
Increasing function
If dy/dx > 0 then y = f(x) is increasing
Decreasing function
If dy/dx < 0 then y = f(x) is decreasing
Monotonically increasing function
y = f(x) is monotonically increasing function if dy/dx>0 ∀ x∈R
Monotonically decreasing function
y = f(x) is monotonically decreasing function if dy/dx<0 ∀ x∈R
Stationary points
When f(x) is neither decreasing or increasing or dy/dx = 0
Or
At the points of contact where tangent is parallel to x-axis then these points are called stationary points or critical points.
Another name of stationary points
Critical points
How do stationary points affect intervals of increase and decrease in a function?
Stationary points separate the interval of increasing and the interval of decreasing.
What type of brackets are used for intervals of increasing or decreasing functions, and why?
Open brackets are used for intervals of increasing or decreasing functions (e.g., (a,b)) because the stationary points themselves are not included in these intervals.
Wavy curve method
The wavy curve method helps determine intervals where a function is increasing or decreasing by analyzing the sign of its derivative. First, find the critical points where the derivative equals zero. Then, plot these points on a number line. Test the sign of the derivative in the intervals between these points. If the derivative is positive in an interval, the function is increasing there; if negative, the function is decreasing. The sign alternates between intervals based on the wavy curve.
Definition of local maxima
When the slope / derivative is moving from +ve to -ve, at the stationary point the function will have local maxima.
Definition of local minima
When the slope / derivative is moving from -ve to +ve, at the stationary point the function will have local maxima.
Term use for minima/maxima
Extrema
Plural of extrema
Extremum
Extremum of degree n
(n-1) bends
∴ (n-1) extremum
Local maxima (mathematically)
y’=0 and y’‘<0
Local minima (mathematically)
y’=0 and y’‘>0
Concavity
Point of inflection (definition)
Point of inflection occur where the concavity of curve changes i.e. where f’‘(x) = 0 and sign changes.
How to find points of inflection
Point of inflection (Mathematically)
Point of inflection in terms of concavity
Saddle point
Odd / even derivative and maxima relationship
Odd / even derivative and minima relationship
