Show that f(x)=x^3-2 x^2+2 x is an increasing function. Hint: Find the minimum value of f^'(x)

Name: Problem 55, Chapter 4: Calculus for AP
Uploaded: 2020-02-13T05:38:17-08:00
Duration: 3 min 16 s
Channel: Linh Vu
Description: Show that f(x)=x^3-2 x^2+2 x is an increasing function. Hint: Find the minimum value of f^'(x)

step 1 So in this question we need to show that the fx is an increasing function everywhere In order to

Show that f(x)=x^3-2 x^2+2 x is an increasing function....

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

First Derivative Test for Monotonicity

This test uses the sign of the derivative to determine whether a function is increasing or decreasing. If a function’s derivative is nonnegative (or positive) over its entire domain, then the function is considered to be increasing, since the slope never falls below zero.

Derivative

The derivative of a function represents the instantaneous rate of change of the function with respect to its variable. It provides information about the slope of the tangent line to the function at any given point and is essential in analyzing the behavior of functions.

Critical Points and Extreme Values

Critical points, where the derivative is zero or undefined, help in identifying local extrema. In the context of showing that a function is increasing, finding the minimum value of the derivative (ensuring it is nonnegative) is crucial, as it confirms there is no interval where the rate of change becomes negative.

Polynomial Function Analysis

Polynomial functions, due to their smooth and continuous nature, are especially suitable for derivative analysis. Their derivatives are also polynomials, which can be easily examined to determine monotonicity and other behaviors across the domain.

Transcript

Please give Ace some feedback