Question

For the special cubic equation $f(x)=a x^{3}+b x+c,$ show that $f(x)$ is always increasing if both $a$ and $b$ are positive and that $f(x)$ is always decreasing if both $a$ and $b$ are negative.

Name: Problem 43, Chapter 5: Calculus Applications and Technology
Uploaded: 2019-06-21T17:41:29-08:00
Duration: 2 min 37 s
Channel: Michelle Ling
Description: For the special cubic equation f(x)=a x^3+b x+c, show that f(x) is always increasing if both a and b are positive and that f(x) is always decreasing if both a and b are negative.

   For the special cubic equation $f(x)=a x^{3}+b x+c,$ show that $f(x)$ is always increasing if both $a$ and $b$ are positive and that $f(x)$ is always decreasing if both $a$ and $b$ are negative.

Calculus Applications and Technology

Edmond C. Tomastik 3rd Edition

Chapter 5, Problem 43

Instant Answer

Solved by Expert Michelle Ling

06/21/2019

Step 1

The derivative of a function tells us the slope of the tangent line at any point on the function. If the derivative is positive, the function is increasing, and if the derivative is negative, the function is decreasing. The derivative of $f(x) = ax^3 + bx + c$ Show more…

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For the special cubic equation $f(x)=a x^{3}+b x+c,$ show that $f(x)$ is always increasing if both $a$ and $b$ are positive and that $f(x)$ is always decreasing if both $a$ and $b$ are negative.

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Key Concepts

Monotonicity

Monotonicity refers to a function's consistent behavior in terms of increasing or decreasing over its entire domain or over specific intervals. When a function’s derivative is always positive, the function is strictly increasing; when it’s always negative, the function is strictly decreasing.

Polynomial Functions

Polynomial functions are expressions involving sums of powers of a variable, with coefficients that influence their shape and behavior. In the context of cubic functions, analyzing their derivatives helps in understanding how the coefficients determine the function's increasing or decreasing nature.

Differentiation

Differentiation is a fundamental tool in calculus that measures how a function changes as its input changes. It provides the rate of change (or slope) at any point, which is crucial for analyzing the behavior of functions, such as determining intervals where they are increasing or decreasing.

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