Types Of Functions

Given,

f(x)%3D(2%20k%2B1)%20x-3-k%20e%5E%7B-x%7D%2B2%20e%5E%7Bx%7D

Since,

f(x)

is monotonically increasing for all

x%20%5Cin%20R%20.

So,

f'(x)%20%5Cgeq%200

(2%20k%2B1)%2Bk%20e%5E%7B-x%7D%2B2%20e%5E%7Bx%7D%20%5Cgeq%200

%5CRightarrow%20e%5E%7B-x%7D%5Cleft((2%20k%2B1)%20e%5E%7Bx%7D%2Bk%2B2%20e%5E%7B2%20x%7D%5Cright)%20%5Cgeq%200

(2%20k%2B1)%20e%5E%7Bx%7D%2Bk%2B2%20e%5E%7B2%20x%7D%20%5Cgeq%200

%5CRightarrow%202%20e%5E%7Bx%7D%5Cleft(e%5E%7Bx%7D%2Bk%5Cright)%2B%20l%20%5Cleft(e%5E%7Bx%7D%2Bk%5Cright)%20%5Cgeq%200

%5CRightarrow%20%5Cleft(2%20e%5E%7Bx%7D%2B1%5Cright)%5Cleft(e%5E%7Bx%7D%2Bk%5Cright)%20%5Cgeq%200

%5CRightarrow%20e%5E%7Bx%7D%2Bk%20%5Cgeq%200

k%20%5Cgeq%200

Hence, least value of

k

is zero.

Chapter Questions
1 MCQs