Maxima And Minima

Given,

y%20%3D2%20%5Ctan%20x-%5Ctan%20%5E%7B2%7D%20x%5C%2C...(i)

%5Ctherefore%20%5C%2C%20%5Cfrac%7Bd%20y%7D%7Bd%20x%7D%20%3D2%20%5Csec%20%5E%7B2%7D%20%5C%2Cx-2%20%5Ctan%20x%20%5Ccdot%20%5Csec%20%5E%7B2%7D%20%5C%2Cx

%3D2%20%5Csec%20%5E%7B2%7D%20x(1-%5Ctan%20x)%5C%2C...(ii)

At point of maxima,

%5Cfrac%7Bd%20y%7D%7Bd%20x%7D%3D0

2%20%5Csec%20%5E%7B2%7D%20x(1-%5Ctan%20x)%3D0

[From E (ii)]

%5Ctherefore%20%5C%2C%20x%3D%5Cfrac%7B%5Cpi%7D%7B4%7D%2C%20%5Cfrac%7B%5Cpi%7D%7B2%7D

Here,

x%3D%5Cfrac%7B%5Cpi%7D%7B2%7D

is not possible]

%5Ctherefore%5C%2C%20x%3D%5Cfrac%7B%5Cpi%7D%7B4%7D

%5B%5Cbecause%20x%20%5Cin%5Cleft%5B0%2C%20%5Cfrac%7B%5Cpi%7D%7B2%7D%5Cright%5D

(given)
Now,

%5Cfrac%7Bd%5E%7B2%7D%20y%7D%7Bd%20x%5E%7B2%7D%7D%3D4%20%5Csec%20x%20%5Ccdot%20%5Csec%20x%20%5Ccdot%20%5Ctan%20x(1-%5Ctan%20x)%20%2B2%20%5Csec%20%5E%7B2%7D%20x%5Cleft(0-%5Csec%20%5E%7B2%7D%20x%5Cright)

%3D4%20%5Csec%20%5E%7B2%7D%20%5C%2Cx%20%5Ctan%5C%2C%20x-4%20%5Csec%20%5E%7B2%7D%20x%20%5Ctan%20%5E%7B2%7D%20x-2%20%5Csec%20%5E%7B4%7D%20%5C%2Cx

%5Cleft.%5Ctherefore%20%5Cfrac%7Bd%5E%7B2%7D%20y%7D%7Bd%20x%5E%7B2%7D%7D%5Cright%7C_%7Bx-%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%3D%204%20%5Csec%20%5E%7B2%7D%20%5Cfrac%7B%5Cpi%7D%7B4%7D%20%5Ctan%20%5Cfrac%7B%5Cpi%7D%7B4%7D-4%20%5Csec%20%5E%7B2%7D%20%5Cfrac%7B%5Cpi%7D%7B4%7D

%5Ctan%20%5E%7B2%7D%20%5Cfrac%7B%5Cpi%7D%7B4%7D-2%20%5Csec%20%5E%7B4%7D%20%5Cfrac%7B%5Cpi%7D%7B4%7D

%3D%204(%5Csqrt%7B2%7D)%5E%7B2%7D%20%5Ccdot%201-4(%5Csqrt%7B2%7D)%5E%7B2%7D%20%5Ccdot(1)%5E%7B2%7D-2%20%5Ccdot(%5Csqrt%7B2%7D)%5E%7B4%7D

%3D%208-8-8

%3D-8

which is negative.

%5Ctherefore

x%3D%5Cfrac%7B%5Cpi%7D%7B4%7D

, function

y%3D2%20%5Ctan%20x-%5Ctan%20%5E%7B2%7D%20x

has maximum value.

%5Ctherefore

Maximum value of function at point

x%3D%5Cfrac%7B%5Cpi%7D%7B4%7D

, will be

%5By%5D_%7Bx-%5Cfrac%7B%5Cpi%7D%7B4%7D%7D%3D1

Given,

y%3D%5Cint_%7B0%7D%5E%7Bx%7D%5Cleft(t%5E%7B2%7D-3%20t%2B2%5Cright)%20dt%5C%2C...(i)

On differentiating w.r.t. '

x%2C

, we get

%5Cfrac%7Bd%20y%7D%7Bd%20x%7D%3Dx%5E%7B2%7D-3%20x%2B2%5C%2C...(ii)

Again, on differentiating w.r.t. '

x'

, we get

%5Cfrac%7Bd%5E%7B2%7D%20y%7D%7Bd%20x%5E%7B2%7D%7D%3D2%20x-3%5C%2C...(iii)

We know that, at point of inflection

%5Cfrac%7Bd%5E%7B2%7D%20y%7D%7Bd%20x%5E%7B2%7D%7D%3D0

%5Ctherefore

From E (iii), we get

2%20x-3%3D0

%5CRightarrow%20x%3D%5Cfrac%7B3%7D%7B2%7D

Now, we have to check behaviour of

%5Cfrac%7Bd%5E%7B2%7D%20y%7D%7Bd%20x%5E%7B2%7D%7D

at point

x%3D%5Cfrac%7B3%7D%7B2%7D

x%3D%5Cfrac%7B3%7D%7B2%7D

Clearly, at

x%3D%5Cfrac%7B3%7D%7B2%7D

sign at

%5Cfrac%7Bd%5E%7B2%7D%20y%7D%7Bd%20x%5E%7B2%7D%7D

changes

%5Ctherefore%5Cleft(%5Cfrac%7B3%7D%7B2%7D%2C%20%5Cfrac%7B3%7D%7B4%7D%5Cright)

is point of inflection

About Maxima And Minima - UPSEE

Maxima And Minima is a vital chapter for UPSEE aspirants. Mastering the concepts covered in this chapter is essential for securing a top rank.

By rigorously practicing the previous year questions associated with this chapter, you can identify high-yield topics, understand the examiner's perspective, and boost your confidence during the actual exam.

Frequently Asked Questions

Why focus on Maxima And Minima PYQs?

Analyzing PYQs for this specific chapter reveals the most frequently tested concepts and the typical complexity of questions, allowing you to tailor your study plan efficiently.

How to best use this analysis?

Review the topic breakdown to see which sub-topics within Maxima And Minima carry the most weight. Then, tackle the questions iteratively to solidify your understanding.

High-Yield Trend

Chapter Questions
2 MCQs

Official Solution

Official Solution

About Maxima And Minima - UPSEE

Frequently Asked Questions

Why focus on Maxima And Minima PYQs?

How to best use this analysis?

Maxima And Minima

High-Yield Trend

Chapter Questions 2 MCQs

Official Solution

Official Solution

About Maxima And Minima - UPSEE

Frequently Asked Questions

Why focus on Maxima And Minima PYQs?

How to best use this analysis?

Chapter Questions
2 MCQs