10. Derivatives and Tangent Lines

d. The First Derivative Test for Maxima and Minima

Local Maxima, Local Minima and Horizontal Inflection Points

A local maximum is a point where the function value is larger than those at nearby points. At a local maximum, the curve changes from increasing to decreasing. In other words, the derivative, or slope, switches from positive to negative.

A local minimum is a point where the function value is smaller than those at nearby points. At a local minimum, the curve changes from decreasing to increasing. In other words, the derivative, or slope, switches from negative to positive.

A local maximum or local minimum is also called a local extremum.

def_max_-x^2

def_min_x^2

At a local minimum or maximum, \(x=c\), the derivative changes sign and so \(f'(c)=0\). However, it is possible to have \(f'(c)=0\) but \(x=c\) is neither a maximum nor minimum. \(f'(c)=0\) means the curve is horizontal at \(x=c\), but it could still be increasing on both sides or decreasing on both sides as shown in the plots.

A horizontal inflection point is a point \(x=c\) where \(f'(c)=0\) and the function is either increasing on both sides of \(x=c\) or decreasing on both sides of \(x=c\) as shown.

There are also inflection points which are not horizontal and we will learn about them in the chapter on Higher Derivatives

def_horinfl_incr

def_horinfl_decr

The First Derivative Test

To summarize:

Suppose \(x=c\) is a critical point of \(f(x)\) where \(f'(c)=0\). Then

If \(f'(x) < 0\) for \(x < c\) and \(f'(x) > 0\) for \(x > c\), then the critical point \(c\) is a local minimum.
This says \(f\) is decreasing on the left and increasing on the right of \(c\).
If \(f'(x) > 0\) for \(x < c\) and \(f'(x) < 0\) for \(x > c\), then the critical point \(c\) is a local maximum.
This says \(f\) is increasing on the left and decreasing on the right of \(c\).
If \(f'(x) < 0\) for \(x < c\) and \(x > c\) or \(f'(x) > 0\) for \(x < c\) and \(x > c\), then the critical point \(c\) is a horizontal inflection point.
This says \(f\) is decreasing on the left and right of \(c\) or increasing on the left and right of \(c\).

When we specify the sign of \(f'\) on \(x < c\) and \(x > c\), we actually mean that there is some interval around \(c\) within which these conditions on \(f'\) hold. For local extrema, we don't care what happens far away from \(c\).

Find the local maxima, local minima and horizontal inflection points of the graph of \(f(x)=x^4-8x^2\).

In an example on a previous page we found the derivative and factored it: \[ f'(x)=4x^3-16x=4x(x-2)(x+2) \] Then we drew a number line and marked where the derivative is positive and negative: eg_number_line-202arrows

By the First Derivative Test, \(f(x)\) has a local minimum at \(x=-2\) where the value is \[ f(-2)=(-2)^4-8(-2)^2=-16 \] \(f(x)\) has a local maximum at \(x=0\) where the value is \[ f(0)=(0)^4-8(0)^2=0 \] and \(f(x)\) has a local minimum at \(x=2\) where the value is \[ f(2)=(2)^4-8(2)^2=-16 \] Its plot is shown.

Find the local maxima, local minima and horizontal inflection points of the graph of \(f(x)=3x^4-16x^3+24x^2\). This graph was previously discussed in an exercise on a previous page.

Factor the derivative and determine where it is positive and negative.

\(f(x)\) is decreasing on \((-\infty,0]\) and is increasing on \([0,\infty)\).

In an exercise on a previous page we found the derivative and factored it: \[ f'(x)=12x^3-48x^2+48x=12x(x-2)^2 \] Then we drew a number line and marked where the derivative is positive and negative: ex_number_line02arrows

By the First Derivative Test, \(f(x)\) has a local minimum at \(x=0\) where the value is \[ f(0)=3(0)^4-16(0)^3+24(0)^2=0 \] and \(f(x)\) has a horizontal inflection point at \(x=2\) where the value is \[ f(2)=3(2)^4-16(2)^3+24(2)^2=16 \] Here is the plot.

10. Derivatives and Tangent Lines

d. The First Derivative Test for Maxima and Minima

Local Maxima, Local Minima and Horizontal Inflection Points

The First Derivative Test

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