Increasing/ Decreasing function

Question 1

Increasing function

Answer 1

If dy/dx > 0 then y = f(x) is increasing

Question 2

Decreasing function

Answer 2

If dy/dx < 0 then y = f(x) is decreasing

Question 3

Monotonically increasing function

Answer 3

y = f(x) is monotonically increasing function if dy/dx>0 ∀ x∈R

Question 4

Monotonically decreasing function

Answer 4

y = f(x) is monotonically decreasing function if dy/dx<0 ∀ x∈R

Question 5

Stationary points

Answer 5

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.

Question 6

Another name of stationary points

Answer 6

Critical points

Question 7

How do stationary points affect intervals of increase and decrease in a function?

Answer 7

Stationary points separate the interval of increasing and the interval of decreasing.

Question 8

What type of brackets are used for intervals of increasing or decreasing functions, and why?

Answer 8

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.

Question 9

Wavy curve method

Answer 9

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.

Question 10

Definition of local maxima

Answer 10

When the slope / derivative is moving from +ve to -ve, at the stationary point the function will have local maxima.

Question 11

Definition of local minima

Answer 11

When the slope / derivative is moving from -ve to +ve, at the stationary point the function will have local maxima.

Question 12

Term use for minima/maxima

Answer 12

Extrema

Question 13

Plural of extrema

Answer 13

Extremum

Question 14

Extremum of degree n

Answer 14

(n-1) bends
∴ (n-1) extremum

Question 15

Local maxima (mathematically)

Answer 15

y’=0 and y’‘<0

Question 16

Local minima (mathematically)

Answer 16

y’=0 and y’‘>0

Question 17

Concavity

Answer 17

Question 18

Point of inflection (definition)

Answer 18

Point of inflection occur where the concavity of curve changes i.e. where f’‘(x) = 0 and sign changes.

Question 19

How to find points of inflection

Answer 19

Question 20

Point of inflection (Mathematically)

Answer 20

Question 21

Point of inflection in terms of concavity

Answer 21

Question 22

Saddle point

Answer 22

Question 23

Odd / even derivative and maxima relationship

Answer 23

Question 24

Odd / even derivative and minima relationship

Answer 24

Question 25

Odd / even derivative and point of inflection relationship

Answer 25

Question 26

Another name of global maxima

Answer 26

Absolute maxima

Question 27

Another name of global minima

Answer 27

Absolute minima

Question 28

Definition of global maxima

Answer 28

The greatest among all the local maxima

Question 29

Definition of global minima

Answer 29

The greatest among all the local minima

Question 30

How do you find the maximum or minimum of functions like y = xᶠ⁽ˣ⁾

Answer 30