# Numbers | Maths with Lemon

Source: https://mathswithlemon.io/aa-calculus-monotony.html


## Monotony

"What is mathematics? It is only a systematic effort of solving puzzles posed by nature." Shakuntala Devi


#### Monotony (Min-Max)


##### What you have to know:

- How to differentiate all functions (If not, check this lesson)

##### Key Points

- 1. Watch the video:
- 2. Check out this presentation .
- 3. Watch this video on increasing and decreasing functions .
- 4. Watch this video on finding the minimum and maximum of a function .

#### Coronavirus and Inflection points


##### Another way of seeing the inflection point and concavity

- In January 2020 there was an outbreak of the coronavirus. The whole world was going crazy that year. The deaths from the virus were rising up to some point and then started decreasing. The whole world was searching for the inflection point of the graph of the death rate.
- 1. Watch the video:

#### Concavity / Inflection Points


##### What you have to know:

- The derivatives of all known functions (If not, check this lesson)
- Find the monotony of a function (If not, check this lesson)
- You must be able to manipulate exponential expressions. (If not, check this lesson)
- You must be able to manipulate logarithmic expressions. (If not, check this lesson)

##### Key Points

- 1. Watch the video:
- 2. Check out this presentation .
- 3. Watch this video .

#### IB questions on inflection points, minimum and maximum


##### What you have to know:

- To find minimum or maximum of a function, take the first derivative equal to zero.
- To find the inflection points, take the second derivative equal to zero and check the concavity.

##### Key Points

- 1. Watch the video:

#### Extra


##### Material and references:

- Hodder Book SL(ISBN: 9781510462359) : 20D, 20E, 20F

##### Key Points

- 1. How to interpret the second derivative of a function.
- A function \( f(x) \) is: — concave-up where \( f''(x) > 0 \) — concave-down where \( f''(x) < 0 \)
- 2. How to find local minimum and maximum points on a graph.
- At a local maximum or minimum \(f'(x) = 0\)
- Given \( f'(a) = 0 \) then if — \( f''(a) < 0 \) then there is a local maximum at \( x = a \) — \( f''(a) > 0 \) then there is a local minimum at \( x = a \) — \( f''(a) = 0 \) then no conclusion can be drawn

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