IIT JEE , AIEEE Forum - Mathematics , Differential Calculus

vishal90

1.. which of the following functions is not monotonic?

(a) e^x- e^-x ( b) e^x+ e^-x (c) e^-1/x

i know how to find mon. inc or dec but how do we check monotonicity

2.. find intervals of monotonicity of the fn f(x) = 2sinx + cosx [ 0, 2pi ]

i dont how to find the intervals-- do u do it with graphs?

krishna.gopal

Monotonic means sign of f'(x) does not changes. Derivative (b) can be both positive and negative so its not monotonic

joyfrancis

The second question becomes pretty confusing when you try to analyse the values so it is better to make the graph.

see,

y=2sinx+cosx

multiply and divide by

5

y=[2sinx/

5 + cosx/

5]

5

Let cos2/

5 = a

y=sin(a+x)

To make the graph of sin(a+x) make the graph of sinx and shift it backwards by a distance of cos inverse 2/

5.

Now you can get the intervals of monotonocity of the function.

I am getting

Increasing --> (-a,90-a)U(135-a,45+a))

Decreasing-->(90-a,135-a)

Correct me if i'm wrong!

spideyunlimited

in Q1) diff of a is e^x + e^-x which will always be positive...
diff of b is e^x - e^-x which can be both positive as well as negative

diff of c is e/ x^2.. also always positive

thus, b is non monotonic ... did u know monotonic is derived from monotonous which means of the same type.

Ans 2) 2 sinx + cos x
f'(x) = 2cosx - sinx
put eqn = 0 .
we get critical point as x = tan inverse 2

and as interval is from 0 to 2 pi
intervals of monotonicity are
x E (0 , tan^-1 2) U ( tan^-1 2 , 2 pi)

PS. plz rate

vishal90

hey joy francis
thanks a lot but the ans given is
inc in (0, 30), (90,150) (270,360)
dec in (30,90) (150,270)

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