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inflection points

Understanding Derivatives, Relative Extrema, and Points of Inflection

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The Big Idea

When studying a function, we often want to answer three important questions:

  1. Is the function increasing or decreasing?
  2. Does the function have any local maxima or minima?
  3. Where does the graph change its shape or curvature?

These questions can be answered using the first and second derivatives.

The first derivative helps us understand the slope of a graph, while the second derivative helps us understand its concavity.

In this post, we will use the function:

f(x)=x^4-2x^3

to understand these concepts.

Understanding the First Derivative

What Is the First Derivative?

The first derivative measures the rate at which a function changes.

It tells us whether the graph is moving upward or downward as we move from left to right.

The first derivative is written as:

f'(x)=\frac{d}{dx}f(x)

Key Interpretation

If:

f'(x)>0

the function is increasing.

If:

f'(x)<0

the function is decreasing.

If:

f'(x)=0

the graph has a horizontal tangent and may contain a maximum or minimum.

Example

Consider the function:

f(x)=x^4-2x^3

Differentiating gives:

f'(x)=4x^3-6x^2

Factoring:

f'(x)=2x^2(2x-3)

To find important points on the graph, we set the derivative equal to zero:

2x^2(2x-3)=0

This gives:

x=0,\quad x=\frac{3}{2}

These values are called critical points.

Understanding Critical Points

What Is a Critical Point?

A critical point is a point where:

f'(x)=0

or where the derivative does not exist.

Critical points are important because maxima and minima can only occur at these points.

Important Note

Not every critical point is a maximum or minimum.

To determine whether a critical point is actually an extremum, we must check how the derivative behaves around it.

Understanding Relative Extrema

What Are Relative Extrema?

Relative extrema are local maximum and minimum points.

A relative maximum occurs when the graph rises and then falls.

A relative minimum occurs when the graph falls and then rises.

The First Derivative Test

We determine extrema by checking whether the derivative changes sign.

Relative Maximum

If:

+\rightarrow-

the function changes from increasing to decreasing.

Therefore, a relative maximum occurs.

Relative Minimum

If:

-\rightarrow+

the function changes from decreasing to increasing.

Therefore, a relative minimum occurs.

Applying the Test

For:

f'(x)=2x^2(2x-3)

we examine the intervals around the critical points.

For values less than 0:

f'(x)<0

For values between 0 and:

\frac{3}{2}

the derivative remains negative.

For values greater than:

\frac{3}{2}

the derivative becomes positive.

This gives the sign pattern:

-\rightarrow-\rightarrow+

Notice that at:

x=0

the derivative does not change sign.

Therefore:

x=0

is not an extremum.

At:

x=\frac{3}{2}

the derivative changes from negative to positive.

Therefore:

x=\frac{3}{2}

is a relative minimum.

Takeaway

A critical point becomes a relative extremum only when the derivative changes sign.

For this function, there is exactly one relative extremum.

Understanding the Second Derivative

What Is the Second Derivative?

The second derivative measures how the slope is changing.

It helps us understand the curvature of a graph.

The second derivative is written as:

f''(x)=\frac{d^2}{dx^2}f(x)

Key Interpretation

If:

f''(x)>0

the graph is concave upward.

If:

f''(x)<0

the graph is concave downward.

Understanding Concavity

Concave Up

A graph is concave upward when it bends like a bowl.

Mathematically:

f''(x)>0

Concave Down

A graph is concave downward when it bends like an upside-down bowl.

Mathematically:

f''(x)<0

The second derivative tells us which type of curvature the graph has.

Understanding Points of Inflection

What Is a Point of Inflection?

A point of inflection is a point where the graph changes concavity.

In other words, the graph changes from:

  • Concave up to concave down

or

  • Concave down to concave up

Key Statement

A point of inflection occurs when the second derivative changes sign.

Important Note

A point where:

f''(x)=0

is only a candidate for an inflection point.

The sign of the second derivative must actually change.

Finding the Points of Inflection

For the function:

f(x)=x^4-2x^3

the second derivative is:

f''(x)=12x^2-12x

Factoring:

f''(x)=12x(x-1)

Setting the second derivative equal to zero:

12x(x-1)=0

gives:

x=0,\quad x=1

These are possible inflection points.

Sign Test

For values less than 0:

f''(x)>0

For values between 0 and 1:

f''(x)<0

For values greater than 1:

f''(x)>0

This gives:

+\rightarrow-\rightarrow+

The sign changes at both points.

Therefore:

x=0

and

x=1

are points of inflection.

Takeaway

A point of inflection occurs whenever the graph changes concavity, which happens when the second derivative changes sign.

Final Results for the Function

For:

f(x)=x^4-2x^3

we found:

  • One relative extremum
  • A relative minimum at
x=\frac{3}{2}
  • Two points of inflection
  • Inflection points at
x=0

and

x=1

Final Takeaway

The first derivative tells us about slope and direction.

\text{First Derivative}\rightarrow\text{Increasing or Decreasing}

The first derivative test helps us identify maxima and minima.

\text{Critical Points}\rightarrow\text{Possible Extrema}

The second derivative tells us about curvature.

\text{Second Derivative}\rightarrow\text{Concavity}

The second derivative test helps us identify points of inflection.

\text{Sign Change in }f''(x)\rightarrow\text{Point of Inflection}

Together, these ideas form the foundation of graph analysis in differential calculus.