SECTION 2 (Maximum Marks: 24)
• This section contains SIX (06) questions.
• Each question has FOUR options. ONE OR MORE THAN ONE of these four option(s) is(are) the correct answer(s).
• For each question, choose the option(s) corresponding to (all) the correct answer(s).
• Answer to each question will be a evaluated according to the following marking scheme :
Full Marks +4 if only (all) the correct option(s) is(are) chosen;
Partial Marks +3 if all the four options are correct but ONLY three options are chosen;
Partial Marks +2 If three or more options are correct but ONLY two options are chosen, both of
which are correct;
Partial Marks + 1 If two or more options are correct but ONLY one option is chosen and it is a
correct option;
Zero Marks 0 if none of the options is chosen (i.e. the question is unanswered);
Negative Marks -2 in all other cases.
Let the function f: R 
R be defined by f(x) = x 3 - x 2 + (x - 1) sin x and let g : R 
R be an arbitrary function. Let fg: R 
R be the product function defined by (fg)(x) = f(x)g(x). Then which of the following statements is/are TRUE?
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
(a, c)
f:R 
R
f(x) = x 3 - x 2 + (x - 1) sin x
g:R 
R
If g is continuous at x = 1 then fg is differentiable
Let h(x) = f(x>g(x)
So, h(x) is differentiable at x = 1
Given h(x) = f(x) . g(x) is differentiable
f'(1) 
0 and g(1) is not define
So, can not comment over continuity and differentiability
Given g(x) is differentiable So, h(x) = f(x) g(x)
h'(x) = f (x) g(x) + g'(x) f(x), as g(x) is differentiable = f(1)g(1) + 0 will exist
Same as for B
can not say about differentiability of the g(x)
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