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International Baccalaureate Mathematics

Calculus

Syllabus Content

Local maximum and minimum points. Testing for maximum and minimum. Optimization. Points of inflexion with zero and non-zero gradients

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Furthermore

This Optimisation video is from Revision Village and is aimed at students taking the IB Maths Standard level course.

Official Guidance, clarification and syllabus links:

Using change of sign of the first derivative or using sign of the second derivative where \(f''(x) \gt 0\) implies a minimum and \(f''(x) \lt 0 \) implies a maximum.

Examples of optimization may include profit, area and volume.

At a point of inflexion, \(f''(x)=0 and changes sign (concavity change), for example \(f''(x)=0\) is not a sufficient condition for a point of inflexion for \(y=x^4\) at (0,0).

Use of the terms “concave-up” for \(f''(x) \gt 0\), and “concave-down” for \(f''(x) \lt 0\).

A local maximum is a point where a function has a higher value than at nearby points, making it a peak in the graph. Conversely, a local minimum is a point where the function has a lower value than its neighbours, forming a trough. To test for these points, we often use the first and second derivative tests. If \( f'(x) \) changes sign from positive to negative as \( x \) increases, \( f(x) \) has a local maximum. If \( f'(x) \) changes from negative to positive, \( f(x) \) has a local minimum. The second derivative test states that if \( f''(x) > 0 \) at a point where \( f'(x) = 0 \), then \( f(x) \) has a local minimum. If \( f''(x) < 0 \), it's a local maximum.

Optimisation involves finding the maximum or minimum values of a function in a given domain, which has numerous applications in real-world scenarios, such as maximising profit or minimising cost.

A point of inflexion is where a curve changes its curvature direction. If the curve changes from concave upwards to concave downwards (or vice versa), it's an inflexion point. At these points, \( f''(x) = 0 \) or is undefined. However, not all points where \( f''(x) = 0 \) are inflexion points. The gradient at an inflexion point can be zero (a horizontal tangent) or non-zero.

Examples:

1) Consider the function \( f(x) = x^3 - 3x^2 \).
The first derivative is \( f'(x) = 3x^2 - 6x \) and the second derivative is \( f''(x) = 6x - 6 \).
Setting \( f'(x) = 0 \), we get \( x = 0 \) and \( x = 2 \). Using the second derivative test, we find that \( f(x) \) has a local maximum at \( x = 0 \) and a local minimum at \( x = 2 \).

2) For the function \( g(x) = x^4 - 4x^3 \),
The first derivative is \( g'(x) = 4x^3 - 12x^2 \) and the second derivative is \( g''(x) = 12x^2 - 24x \).
Setting \( g''(x) = 0 \), we get \( x = 0 \) and \( x = 2 \). Both of these are points of inflexion, but only \( x = 2 \) has a gradient of zero.

A stationary point has a gradient of zero.

A turning point is a minimum or maximum.

A point of inflection is where the second derivative is zero

This video on Optimization and Calculus Curves is from Revision Village and is aimed at students taking the IB Maths AA SL/HL course.

Transum,

Saturday, August 17, 2019

"My favourite optimisation problem is the farmer's dilemma. It's one of the Advanced Starters and can be found here

Fence Optimisation"

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