Suppose the total cost C(x) (in dollars) to manufacture a...

Suppose the total cost $C(x)$ (in dollars) to manufacture a quantity $x$ of weed killer (in hundreds of liters) is given by
$$C(x)=x^{3}-2 x^{2}+8 x+50$$
a. Where is $C(x)$ decreasing?
b. Where is $C(x)$ increasing?

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Key Concepts

Derivative

The derivative of a function is a fundamental concept in calculus that represents the rate at which the function's value changes with respect to a change in its input. It serves as the basis for understanding the behavior of the function, including where it is increasing or decreasing.

Critical Points

Critical points are values in the domain of a function where the derivative is zero or undefined. Identifying these points is essential as they often signal potential changes in the function’s behavior, such as transitioning from increasing to decreasing or vice versa.

Increasing Functions

A function is said to be increasing on an interval if, for any two points in that interval, a larger input produces a larger output. This behavior is confirmed when the derivative of the function is positive over that interval.

Decreasing Functions

A function is decreasing on an interval if, for any two points in that interval, a larger input results in a smaller output. This is indicated by a negative derivative on that interval, which shows that the function's values are falling.

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