KVPY SX SB Exam 2018 Question Paper

Question : 13 of 120

Marks: +1, -0

Let ln(x) denote the logarithm of x with respect to the base e. Let S ⊂ R be the set of all points where thefunction ln

(x^2 - 1)

is well-defined. Then the number of functions f : S → R that are differentiable, satisfy

\f ’(x) = \ln(x^2-1)

for all x ∈ S and f(2) = 0, is

Solution:

x^2 - 1 > 0

x ∈ (–∞, –1) ∪ (1, ∞)

\f’(x) = \ln(x^2 - 1)

=x\ln(x^2-1)-2x-\log{|{x-1}/{x+1}|}+c

(integrating by parts)

\f(x)=\{\table x\ln(x^2-1)-2x-\log({1-x}/{1+x})+c_1,\when x < -1;x\ln(x^2-1)-2x-\log({x-1}/{x+1})+c_2,\when x > 1

While applying initial condition f (2) = 0 we will get only

c_2

but not

c_1

.
hence , infinite function possible.

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