In problems 1-9, determine the concavity and points of inflection of the graph of the function.

In problems 1-9, determine the concavity and points of...

Key Concepts

Differentiation

Differentiation is the process of computing the derivative of a function, which measures how the function changes as its input changes. This process is fundamental in calculus for understanding the behavior of functions, particularly in determining rates of change and preparing for further analysis such as finding higher-order derivatives.

Second Derivative Test

The second derivative test involves computing the second derivative of a function to analyze its concavity and identify points where the curvature might change. If the second derivative is positive on an interval, the function is concave up there; if negative, concave down. This test is essential for determining the local extrema and identifying candidate points for inflection.

Concavity

Concavity describes the curvature of a function's graph. When a function is concave up, the graph bends upward like a cup, implying that the tangent lines lie below the graph. Conversely, when a function is concave down, the graph bends downward, with tangent lines lying above it. Analyzing concavity helps in understanding the overall shape and behavior of the function.

Points of Inflection

Points of inflection are specific points on the graph where the concavity changes from up to down or vice versa. These points occur when the second derivative changes sign, signaling a transition in the curvature of the graph. Identifying points of inflection is crucial for a comprehensive understanding of a function’s graphical behavior.

Transcript

00:01 We are going to determine where this function is concave up and down, as well as where we have inflection points.

00:07 We will need to get to the second derivative.

00:09 The first derivative, we would use the product rule, first times the derivative of the second, which is one, plus the second times the derivative of the first, which would use the chain rule.

00:27 Power out front, subtract one from the old power.

00:31 So now that's squared and then multiply by the derivative of the inside, but that's just one.

00:38 I'm going to rewrite this because we can pull out some common factors.

00:44 There's a common x minus two cube.

00:47 So let's make x minus two squared.

00:49 So let's go x minus two squared out front.

00:53 And then on the inside, the first set we're left with x minus two.

00:58 The second one, we're left with three times x minus one, which distributed is three x minus three.

01:08 So in total, we end up with x minus two squared times four x minus five.

01:18 Second derivative will also use the product rule.

01:24 First, times the derivative of the second, which is four, plus the second.

01:34 Times the derivative of the first, which uses the chain rule.

01:37 Power out front, subtract one from the old power, so that's to the first, and multiply by the derivative of the inside, which is one.

01:47 Again, there is a common factor i can pull out front.

01:51 I'll pull a two times x minus two out front.

01:56 That would leave me here with two times the quantity x minus two, and in the second one we're left with 4x minus 5.

02:10 So we end up with 2 times x minus 2...

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