The standard normal probability function is used to describe...

The standard normal probability function is used to describe many different populations. Its graph is the well known normal curve. This function is defined by
$$f(x)=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2}$$
Give the intervals where the function is increasing and decreasing.

Key Concepts

Exponential Function Properties

Exponential functions, involving the constant e, have unique growth and decay properties that are smooth and continuous. When differentiating functions incorporating exponentials, understanding these properties, along with rules such as the chain rule, is crucial for accurately determining behavior such as rates of increase or decrease.

Increasing and Decreasing Functions

A function is increasing on an interval if, whenever one point is less than another, the function’s output also increases. Conversely, it is decreasing if a larger input leads to a smaller output. These concepts are fundamental in understanding the behavior of functions in calculus and are typically determined by investigating the sign of the derivative.

Derivative and Monotonicity

The first derivative of a function indicates the slope at any given point. Determining where the derivative is positive or negative allows us to identify intervals where the function is increasing or decreasing, respectively. Setting the derivative equal to zero helps locate critical points, which are potential boundaries between increasing and decreasing intervals.

Critical Points

Critical points occur where the derivative is zero or undefined and are pivotal in studying a function’s behavior. In the context of monotonicity, they mark potential transitions between increasing and decreasing behavior, making them essential for partitioning the domain into intervals where the function behaves consistently.

Standard Normal Distribution

The standard normal distribution is a key concept in probability and statistics, characterized by a mean of 0 and a standard deviation of 1. Its probability density function, which has a bell-shaped curve, exhibits symmetry and unique mathematical properties that make it a fundamental model for analyzing random variables in a variety of applications.

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