RD Sharma Class 12 Solutions Chapter 17 - Increasing and Decreasing Functions (Ex 17.1) Exercise 17.1 - Free PDF
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Let us first define what functions are before discussing growing and decreasing functions as well as monotonicity. A function is a relationship between input and output in which each input corresponds to only one output.
For intervals across their whole domain, functions might increase, decrease, or remain constant. In the provided intervals, functions are continuous and differentiable.
In mathematics, an interval is a continuous/connected segment of the real line. It's essentially a line segment with two endpoints because it's a part of a line. As a result, an interval has two ends. To make it easier to keep track, let's name our interval and endpoints, and assume any two points in an interval.
FAQs on RD Sharma Class 12 Solutions Chapter 17 - Increasing and Decreasing Functions (Ex 17.1) Exercise 17.1
1. What does it mean by non - decreasing functions?
When each element is greater than the one before it, it is said to be increasing. When no element is smaller than the one before it, it is said to be non-decreasing. To summarize, each element is either greater than or equal to the one before it. A non-decreasing function is one in which x1 x2 f(x1) f(x2) f(x1) f(x2) f(x1) f(x2) f(x1) f(x2) f(x1) f(x2) f(x1) f(x2) (x2).
To put it another way, take two x-values on an interval. The function is said to be non-decreasing if the function value at the first x-value is less than or equal to the function value at the second. However, this straightforward approach has the drawback of allowing the function to decrease somewhere in between the two positions. Non-decreasing functions must be thought of in terms of derivatives to be properly defined.
2. What is the interval of decrease?
The domain of any given function where its value is proportionately larger or smaller is referred to as the interval of increase and reduction. If y increases as x increases (reading from left to right), the function is said to be rising. In plain English, the graph gets uphill as you look at it from left to right. The slope of this graph is positive. If y drops as x grows (reading from left to right), the function is said to be declining. So, the graph slides downhill from left to right as you look at it. The slope of this graph is negative.
3. What is a monotonic function?
The term monotonicity of the function refers to the growing or decreasing behavior of functions. Any function that follows one of the four situations listed above is called a monotonic function. Monotonic is a term that consists of two components. Mono denotes one, while tonic denotes a tone. It signifies "in one tone" when used together. Does it imply that a function is expanding when we say it is non-decreasing? No. It could also mean that the function does not change at all! In other words, a function with a constant value throughout a period of time. Non-decreasing should never be confused with growing. That was how growing and decreasing functions were defined.
4. What is meant by decreasing functions?
Decreased functions are those that decrease over a particular time frame. The function y = f(x) demonstrates that the function has the interval points x1 and x2 via which it might be decreasing or strictly decreasing at the specified time interval. This inequality of(x1) f(x2) distinguishes it from strictly decreasing functions since the line of the graph remains equal for both axes in a given interval of time. This strictly decreasing function is defined as a function that rigorously holds the inequality and does not have the same value for both axes at a given interval of time, and is represented as f(x1) > f(x2) (x2).
5. What is meant by increasing functions?
An increasing function is a function that increases at a particular interval of time. When the function y = f(x) involves an increasing function with two points of intervals (x1 and x2) and the functional inequality f(x1) f(x2), the function is said to be an increasing function. Some parts of the rising function remain equal with both functions in this inequality, distinguishing it from a strictly increasing function. While a strictly growing function does not get equal to both axes in the middle of the function's increasing phase during the provided time interval. For free study materials, access the Vedantu app and website.