GATE CSE
First time here? Checkout the FAQ!
x
+3 votes
286 views
asked in Calculus by Boss (6.9k points)   | 286 views

2 Answers

+8 votes
Best answer

$e^\sqrt{2}$ will be the highest value of the given expression.

$\frac{e^{\sin x}}{e^{\cos x}}$ can be written as $e^{\sin x - \cos x}$.

$e$ is a famous irrational constant that is used as base of natural logarithms, and its approximate value is $2.7183$, (which is of course greater than $1$).

So now we just have to maximize $(\sin x - \cos x)$ to find the maximum value of given expression.


Clearly $\sin x$ and $\cos x$ are differentiable functions, hence their difference $(\sin x - \cos x)$ is also differentiable.

So we can differentiate $\sin x - \cos x$ to find its maximum value.

Solution of the equation $\frac{d}{dx}(\sin x - \cos x) = 0$ will give us the points where $\sin x - \cos x$ will attain its maximum value.

On differentiating $(\sin x - \cos x)$ with respect to $x$, we get $\cos x + \sin x$.

Putting $\cos x + \sin x = 0$ and simplifying we get 

$\tan x = -1$

This equation has infinitely many solutions.

One of them is $x = \frac{3\pi}{4}$$x = \frac{3\pi}{4} \text{radians or 135 degrees}$ .

$\sin \left(\frac{3\pi}{4} \right ) = \frac{1}{\sqrt2} \text{ and }\cos \left(\frac{3\pi}{4} \right ) = \frac{-1}{\sqrt2}$.

Putting these values in $e^{\sin x - \cos x}$, we get $e^{\sqrt2}$ or $e^{1.414}$.

 

answered by Veteran (12.8k points)  
selected by
0 votes

Say,Y=esinx / ecosx 

=>log Y =log(esinx / ecosx)

=>log Y =log esinx - log ecosx

              =sinx - cosx

           =1

 log Y =log e

      Y=e

answered by Veteran (53.9k points)  
sin x - cos x = 1?

what is the error if x=⊼/2

then sin⊼/2 -cos⊼/2=1 . isnot it?

yes, but that need not be maximum as cos x can go negative. $\frac {1}{\sqrt 2} - (-\frac{1}{\sqrt 2})$ gives the maximum value here.


Top Users May 2017
  1. akash.dinkar12

    3578 Points

  2. pawan kumarln

    2314 Points

  3. Bikram

    1950 Points

  4. Arjun

    1852 Points

  5. sh!va

    1682 Points

  6. Debashish Deka

    1296 Points

  7. Devshree Dubey

    1282 Points

  8. Arunav Khare

    1122 Points

  9. Angkit

    1072 Points

  10. LeenSharma

    1028 Points

Monthly Topper: Rs. 500 gift card
Top Users 2017 May 29 - Jun 04
  1. Arunav Khare

    246 Points

  2. Arjun

    202 Points

  3. pawan kumarln

    108 Points

  4. Rupendra Choudhary

    94 Points

  5. Niharika 1

    90 Points


22,909 questions
29,243 answers
65,403 comments
27,745 users