Differentiation, derivatives and Taylor approximations: Applications of derivatives
Increasing, decreasing, and extreme values of mathematical functions
Increasing, decreasing and stationary functions The behaviour of a function can be investigated using the derivative. The relationship between the graph of a smooth function \(f(x)\) on an interval \(I\) and derivative \(f'(x)\) on the same interval is namely: \[\begin{aligned} \text{the graph of }f(x)\text{ is increasing on }I \iff f'(x)>0\\
\text{the graph of }f(x)\text{ is decreasing on }I \iff f'(x)<0\\
\text{the graph of }f(x)\text{ is stationary on }I \iff f'(x)=0\end{aligned}\]
Extrema \(f(a)\) is a global maximum (minimum) of the function \(f\) when the function takes its largest (smallest) value in \(a\), i.e., \(f(a)\ge f(x)\) for all \(x\) (or in case of a local minimum \(f(a)\le f(x)\)). \(f(a)\) is a local maximum (maximum) of the function \(f\) when the function takes its largest (smallest) value in the neighbourhood of \(a\). The general term for maximum or minimum is extremum (plural: extrema) and the function values are also called extreme values.
If you want to find an extremum of a smooth function, you can first look at a point where the graph is stationary (also referred to as a critical point), that is, a point in the domain at which the tangent line on the graph is horizontal (derivative equal to zero). After all, each extremum is attained at a critical point because the derivative changes sign at this point. Once you have found a critical point, you must still determine whether it corresponds to an extremum and, if so, what kind of extremum it is. The following method applies:
Local extrema via the derivative sign pattern If \(f'(a)=0\) and the sign of \(f'(x)\) changes at \(x=a\) from positive to negative,
then \(f(x)\) has a local maximum at \(x=a\).
If \(f'(a)=0\) and the sign of \(f'(x)\) changes at \(x=a\) from negative to positive,
then \(f(x)\) has a local minimum at \(x=a\) .
The second derivative test for local extrema is a useful alternative.
The second derivative test The function \(f(x)\) has a local maximum at \(x=a\) if \(f'(a)=0\) and \(f''(a)<0\).
The function \(f(x)\) has a local minimum at \(x=a\) if \(f'(a)=0\) and \(f''(a)>0\).
If \(f'(a)=0\) and \(f''(a)=0\), then you cannot conclude what kind of critical point \(x=a\) is: it may correspond with an extremum, but it may also be a so-called inflection point, that is, a point where the derivative attains a maximum or minimum.
For the calculation of inflection points of a function you use the second and third derivative:
Inflection point If \(f''(a)=0\) and \(f'''(a)\neq 0\), then the function \(f(x)\) has an inflection point at \(x=a\).
Mathcentre video
Maxima and Minima (38:19)
