Find all relative extrema. Use the Second Derivative Test where applicable. f(x)=x^2 e^-x

Name: Problem 51, Chapter 4: Calculus Early Transcendental Functions
Uploaded: 2021-07-21T04:48:26-08:00
Duration: 3 min 43 s
Channel: Jonathon Brumley
Description: Find all relative extrema. Use the Second Derivative Test where applicable. f(x)=x^2 e^-x

step 1 We are going to locate our relative extrema using the second derivative test, if possible. So le

Find all relative extrema. Use the Second Derivative Test...

Key Concepts

Relative Extrema

Relative extrema refer to the local maximum and minimum values of a function. These are points where the function changes from increasing to decreasing or vice versa, and are characterized by the function being higher or lower than all nearby points.

Critical Points

Critical points of a function are the values of the independent variable where the first derivative is zero or undefined. These points are important because potential relative extrema occur at these points, and they are used as starting candidates for further analysis.

First Derivative

The first derivative of a function measures its instantaneous rate of change. By finding where the derivative is zero or does not exist, one identifies potential critical points, which are candidates for relative extrema.

Second Derivative Test

The Second Derivative Test is a method used to classify critical points by evaluating the second derivative. If the second derivative at a critical point is positive, the function has a local minimum there; if it is negative, the function has a local maximum; and if it is zero, the test is inconclusive.

Product Rule

The product rule is a fundamental technique for differentiating functions that are the product of two or more simpler functions. It states that the derivative of a product is given by the first function times the derivative of the second plus the second function times the derivative of the first.

Chain Rule

The chain rule is used in differentiation when dealing with composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

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