Find the greatest and least value of the function f(x)=(1)/(sinx+4)+(1)/(cosx-4) for all x in R.

Name: Problem 28, Chapter 9: Differential Calculus Booster with Problems and Solutions for JEE Main and Advanced
Uploaded: 2022-04-14T03:21:30-08:00
Duration: 5 min 59 s
Channel: Monisha Shukla
Description: Find the greatest and least value of the function f(x)=(1)/(sinx+4)+(1)/(cosx-4) for all x in R.

Find the greatest and least value of the function...

Key Concepts

Composite Function Analysis

When different types of functions, like trigonometric and rational functions, are combined, each function’s individual properties interact in ways that influence the overall behavior. Understanding these interactions helps in accurately assessing the composite function’s range and identifying the maximum and minimum values it can achieve.

Optimization in Calculus

The process of finding the greatest and least values of a function relies on optimization principles. Techniques such as taking derivatives to locate critical points or analyzing the behavior of the function at the boundaries of its domain are central to determining extrema, especially when dealing with functions that involve composite forms.

Rational Functions and Domain Restrictions

When functions are written as ratios, the denominator imposes critical restrictions; it must never be zero. In contexts where trigonometric functions appear in the denominator, it becomes necessary to analyze those trigonometric expressions to determine where the function is defined, ensuring a complete and accurate evaluation of the function's behavior.

Transformation of Functions

Transformations, such as adding a constant to a trigonometric function, shift its range and behavior. Recognizing how these shifts affect the overall function is crucial, especially when the transformed trigonometric functions serve as inputs to other operations like reciprocals, altering both the domain and the range.

Trigonometric Function Ranges

Trigonometric functions such as sine and cosine are inherently bounded, typically within [-1, 1]. This understanding is essential when these functions are transformed or used within larger expressions, as it limits the possible values that the function can reach and impacts subsequent calculations involving these functions.

Transcript

00:01 If f of x is equal to 1 by sine x plus 4 plus 1 by cos x minus 4 here so we have to find maximum and minimum value for this function here so f dash x to find maximum or minimum value we will differentiate this f with respect to x and equate this f dash x with equal to 0 here then we will find roots and then we can say function is maximum or minimum or that point or not so f -dash -x, differentiate this f with respect to x, both sides, we can write f -d -x equals to here differentiation of sine x, 1 by sinex plus 4, is equal to 1 by sinex plus 4 whole square.

00:44 And as we know that, differentiation of sinex is equal to cos of x here and here is with minus sine plus.

00:51 Similarly here, differentiation of 1 by cos x minus 4 is equal to 1 by cos x minus 4 whole square with minus sign here.

01:00 We know the differentiation of course of x equals to minus of sine x so plus of sine x which is equals to here we will put f dash x as 0 so this term will going to 0 or by simplifying this we can write it as here course of x divided by sine x plus 4 whole square which is equals to here sine of x divided by course of x minus of 4 whole square here so by simplifying this we will will cross multiply here or we can write this as here course of x into course of x minus 4 whole square which is equal to sine of x into sine x plus 4 whole square here or by simplifying this we can write it as here course of x into this term can be written as here cos square x plus 16 minus of 2 a b 8 of course of x which is equal to here sine x into sine square x plus 16 plus of 8 of sine of x here so by simplifying this we can write it as here cos cube x minus of sine cube x minus of 8 into cos square x minus of sine square x plus of 16 into course of x minus of sine of x which is equal to 0 here or by simplifying this here, cos cube x minus sine cube x.

02:32 Let us take cos x minus of sine x as common from complete equation here.

02:36 So this term can be written as here.

02:39 This term can be written as here.

02:43 Kosso square x plus sine square x minus of sine x.

02:46 Cos x as we know that kosso square theta plus sine square theta equals to 1...

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