Show that for f(x)=x^6 we have f^''(0)=0 but the point x=0 is not an inflection point of f.

Name: Problem 9, Chapter 3: Calculus: Early Transcendentals
Uploaded: 2022-02-24T21:41:00-08:00
Duration: 1 min 23 s
Channel: James Kiss
Description: Show that for f(x)=x^6 we have f^''(0)=0 but the point x=0 is not an inflection point of f.

step 1 If we have the function f of x equals x to the sixth, then the first derivative would be 6x to t

Show that for f(x)=x^6 we have f^''(0)=0 but the point x=0...

Key Concepts

Inflection Point

An inflection point is a point on the curve of a function where the concavity of the function changes. That is, the function changes from being concave up (cup-shaped) to concave down (cap-shaped) or vice versa. Determining an inflection point typically involves checking the sign changes of the second derivative around the point.

Concavity

Concavity refers to the direction in which a curve bends. A function is concave up if its graph lies above its tangent lines (the slopes are increasing) and concave down if its graph lies below its tangent lines (the slopes are decreasing). Assessing concavity is essential in understanding the behavior of a function, especially when evaluating potential inflection points.

Derivative

In calculus, the derivative of a function measures the rate at which the function's value changes with respect to changes in its input. It provides critical information about the function's behavior, such as slopes of tangent lines and points where the function may have local extrema or changes in monotonicity.

Second Derivative

The second derivative is the derivative of the first derivative. It is used to analyze the curvature, or concavity, of the function. A zero second derivative at a point can sometimes indicate the possibility of an inflection point; however, further investigation is needed since a zero second derivative does not guarantee a change in concavity.

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