IIT JEE , AIEEE Forum - Mathematics , Differential Calculus

gauravstud

find the maximum value of (sinx)^cosx +(cosx)^sinx

raulrag009

is it 1

gauravstud

explain it

agnit_thebest

can we do it like this???
let us consider the max value of sin x as 1 then cos x comes as 0.

so we get 1^0 + 0^1 = 1

antonyajay21

amitp91

hsbhatt

Guys, if sinx and cosx are both positive, then sinx^cosx+cosx^sinx is greater than 1.

We have cosx<2. Since sinx<1, sinx^cosx>sin²x

Similarly, cosx^sinx > cos²x

Hence sinx^cosx+cosx^sinx >1 if sinx and cosx are +ve

So, 1 is not the answer.

I will try to get a more rigorous answer. But just look at this.

Choose sinx = 1/10ⁿ

Let cosx be negative. so cosx = -

(1-sin²x) = -

(1-1/10²ⁿ)

So, sinx^cosx+cosx^sinx for very large n becomes 10ⁿ+1

If you choose sinx = -1/10ⁿ with a similar choice of cosx, you would get -10ⁿ+1.

So it is unbounded

priyesh

the function is defined for those x in which both cosx & sinx >0

so analysing the function in the first quadrant

f(x) = sinx^cosx + cosx^sinx

for sum to be max both sinx & cosx should be at thier max but when sinx is 1 cosx is 0 & vice versa so from the nature of sinx & cosx maxima should arise at x = pi/4

so max value = 2* [1/root(2)^1/root(2)]

hsbhatt

Who said anything about a function?

hsbhatt

I seem to have created a small conceptual difficulty because when we come to cosx^sinx we have a negative number being raised to a power that has an even power, something akin to taking square root of a -ve number. But we can still tide over that difficulty.

With our choice of cosx, we get cosx = -m where m is a +ve number which is almost 1.

So, cosx = meⁱ

Now cosx^sinx = (meⁱ)^{1/10^n} = (m)^{1/10^n} [cos(

/10ⁿ) + isin(

/10ⁿ)]

So, in the limit n

cosx^sinx tends to 1.

So, we step into complex numbers and quickly step back into real numbers leaving us still "pure".

swashata4iit

I think 1 is the answer... May be agnit_thebest is rite... check the graph

chetan_kp

well did anybody get the answer
i m getting somewht strange one ie
(5-4

2)/4

Indian_Dragon

1 is not the correct answer .

It is sumthing about 1.57

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