DIFFERENTIATION 14

Optimisation

Optimisation uses derivatives to find the largest or smallest possible value of a quantity.

The main idea

An interior optimum of a differentiable function often occurs where

\[\boxed{f'(x)=0}\]

but critical points and endpoints must also be considered when relevant.

1. Why derivatives help

At a smooth point where a function changes from increasing to decreasing or vice versa, its tangent is horizontal.

2. Typical process

\[\boxed{\text{define variables}\rightarrow\text{form function}\rightarrow\text{differentiate}\rightarrow\text{find candidates}\rightarrow\text{identify optimum}}\]

3. Maximum-area example

A rectangle of perimeter \(20\) satisfies \(x+y=10\), so \(y=10-x\). Its area is

\[A(x)=x(10-x)=10x-x^2.\]

Then \(A'(x)=10-2x\), so \(A'(x)=0\) gives \(x=5\) and \(y=5\).

4. Confirming the maximum

Since \(A''(5)=-2\lt0\), the stationary point is a maximum. The maximum area is \(25\).

5. Minimum example

For \(f(x)=x^2-6x+13\), \(f'(x)=2x-6\) gives \(x=3\). Since \(f''(3)=2\gt0\), the minimum value is \(f(3)=4\).

6. Local and global optima

A local optimum compares nearby values. A global optimum compares all values in the domain under consideration.

7. Endpoints matter

On a closed interval \(a\le x\le b\), compare relevant interior critical points with the endpoint values.

8. Critical points

Possible optima can occur where \(f'(x)=0\) or where the derivative does not exist while the function is defined.

9. Constraints

A constraint often allows several variables to be reduced to one variable before differentiation.

10. Interpretation

A mathematical solution must make sense in context. For a length, for example, \(x\gt0\).

11. Practical method

\[\boxed{\text{model}\rightarrow\text{differentiate}\rightarrow\text{critical points/endpoints}\rightarrow\text{classify}\rightarrow\text{interpret}}\]