# RD Sharma Class 12 Chapter 17 Solutions (Increasing And Decreasing Functions)

RD Sharma Class 12 Maths Solutions for Chapter 17 Increasing and Decreasing Functions talk about the increasing and decreasing functions. These solutions help you to fully prepare for this concept and apply them to solve questions you get in CBSE and several competitive exams. The best part about studying RD Sharma Class 12 Solutions for Maths is that it offers you a chance to level up your knowledge that you get in other books. It is so far the best book to prepare yourself well for all upcoming examinations.

The book talks about various topics such as a solution of rational algebraic inequations, algorithms, strictly increasing functions, strictly decreasing functions, monotonic functions, necessary and sufficient conditions for monotonicity, finding the interval where a given function is decreasing or increasing, etc. These questions are arranged in the same order as you find them in RD Sharma Solutions for Chapter 17 Increasing and Decreasing Functions. The chapter comprises 47 questions in two exercises to help you get more familiar with the type of questions.

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## Topics Discussed in RD Sharma Class 12 Maths Solutions Chapter 17 Increasing and Decreasing Functions

This chapter discusses the monotonicity of function, f(x). f(x) can be called as a monotonically increasing function on the closed interval [a,b] if and only if the value of this function increases as well as decreases when the variable x increases or decreases. When x’s value increases resulting in the decreasing of the value of the function f(x), then f(x) can be called as a monotonically decreasing function. Any function’s monotonicity in the closed interval [a,b] is strongly associated with its derivative in the closed interval [a,b].

**The solution of Rational Algebraic Inequations**

The following results are considered to be very useful in providing the solution of Rational Algebraic Inequations

- ab > 0 => (a > 0 and b > 0) or (a < 0 and b < 0)
- ab < 0 => (a > 0 and b < 0) or (a < 0 and b > 0)
- ab > 0 and a > 0 => b > 0
- ab < 0 and a < 0 => b > 0

Let A(x) and B(x) ar the two polynomials, then the following Inequations

{A(x) / B(x)} > 0,

{A(x) / B(x) } ≥ 0

{A(x) / B(x)} < 0

And {A(x) / B(x)} ≤ 0

Will be known as rational algebraic inequations.

**Algorithm**

- Factorize into linear factors the two polynomial functions, A(x) and B(x)
- Keep x’s coefficient having a positive value in all the factors.
- Equating each factor to 0 and finding the x’s corresponding values. You may call these values as critical points.
- On the number line, plot all the critical points. The ‘n’ critical points divide the line in (n+1) regions.
- You’ll see in the rightmost region, the expression will always be positive. However, in regions other than the rightmost region, it can be found as alternatively negative and positive. You may mark positive signs again in the rightmost region. Then the negative and positive signs can be marked alternatively in the rest of the regions.
- For the given inequation, get the solutions set by selecting all the appropriate regions in the aforementioned step.

**Strictly increasing functions **

Call a function f(x) a strictly increasing function on an open interval (a,b) if the following holds true.

x_{1} < x_{2} => f(x_{1}) < f(x_{2}) for all values of x_{1}, x_{2} ϵ (a, b)

Therefore, f(x) will be strictly increasing on (a,b) if the f(x)’s value increases with increase in x.

**Strictly decreasing functions**

Call f(x), function a strictly decreasing function on an open interval (a,b) if

x_{1} < x_{2} => f(x_{1}) > f(x_{2}) for all values of x_{1}, x_{2} ϵ (a,b)

So you may call f(x) strictly decreasing on an open interval (a,b) if f(x)’s values decrease with the rise in x’s values.

### Discussion of exercises in RD Sharma Class 12 Maths Solutions Chapter 17 Increasing and Decreasing Functions

- Exercise 17.1 may ask you to prove a given function is increasing or decreasing or neither on a given interval.
- Exercise 17.2 asks you to find the intervals for given functions if it is increasing or decreasing. You may also be asked to show whether a given is increasing or decreasing.

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