Concavity Determine the intervals on which the following functions are concave up or concave dow

Concavity Determine the intervals on which the following...

Key Concepts

Concavity

Concavity refers to the direction in which a function curves. A function is said to be concave up if its graph is shaped like a cup (the tangent lines lie below the graph), and concave down if it is shaped like a cap (the tangent lines lie above the graph). This characteristic helps in understanding the behavior of the function and in identifying local minima and maxima.

Second Derivative

The second derivative of a function provides information about the function's concavity. By analyzing the sign of the second derivative, one can determine whether a function is concave up (second derivative positive) or concave down (second derivative negative) on a particular interval. It is a crucial step in understanding the overall shape of the graph.

Inflection Points

An inflection point is a point on the graph of a function where the concavity changes, that is, the function changes from being concave up to concave down or vice versa. Identifying an inflection point typically involves finding where the second derivative is zero or undefined and confirming that the sign of the second derivative actually changes around that point.

Differentiation Techniques

Differentiation techniques, such as applying the product rule and chain rule, are essential for finding first and second derivatives of functions, especially when the function is composed of multiple parts. These methods help break down complex functions into simpler pieces, enabling the calculation of derivatives needed for analysis of the function’s concavity and inflection points.

Transcript

00:01 We are starting with a function of e to the x times x minus 3, and we are determining where this is concave up, concave down, as well as the inflection points.

00:11 To look at concavity, we need to get to the second derivative.

00:16 The first derivative is going to use the product rule.

00:20 We have first term times the derivative of the second, which is one, plus the second term, which is the x minus 3, times the derivative.

00:32 Of the first, which is e to the x.

00:35 And i can simplify that down some.

00:37 That would give me e to the x plus x e to the x minus 3e to the x.

00:47 And i can combine the like terms of e to the x minus 3e to the x.

00:52 So i really have negative 2e to the x plus x times e to the x.

01:01 My second derivative, derivative of the negative 2e to the x is negative 2, e to the x, and the derivative of the x, e to the x will use the product rule.

01:18 First times the derivative of the second, which is e to the x, plus the second term e to the x, times the derivative of the first, which is 1.

01:29 We will get possible inflection points any time the second derivative is non -differentialable, or it is equal to 0.

01:38 Is never non -differentiable, so let's just check to see where the second derivative is equal to zero.

01:45 And i'll call this a pip for possible inflection points.

01:50 I can simplify it down a little bit more to if i want to, because negative 2e to the x plus e to the x is the same as negative e to the x.

02:02 Okay, so let's see where this equals zero.

02:10 I can factor out an e to the x, so that would give me e to the x...

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