SECTION - 2 : (Paragraph Type)
This section contains 6 multiple choice questions relating to three paragraphs with two questions on each paragraph.
Each question has four choices
R (the set of all real numbers) be a function. Suppose the function f is twice differentiable, f(0) =f(1) = 0 and satisfies f ” (x) - 2f ’ (x) + f(x) 
e x , x 
[0, 1]. (i) Which of the following is true for 0 < x < 1 ? 
(ii) If the function e -x f(x) assumes its minimum in the interval [0, 1] at x = 
, which of the following is true ? 
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
(i)
Let g (x) = e – x f (x)
and g ” (x) > 1 > 0
So, g (x) is concave upward and g (0) = g (1) = 0
Hence, g (x) < 0 
x 
(0, 1)
e – x f (x) < 0
f (x) < 0 
x 
(0, 1)
Alternate Solution
f"(x) – 2f ’ (x) + f (x) 
e x
Let g (x) = f (x) e – x – 
g (0) = 0, g (1) = 
Since g is concave up so it will always lie below the chord joining the extremities which is y = 
(ii)
Let, g (x) = e – x f (x)
As g ” (x) > 0 so g ’ (x) is increasing.
So, for x < 1/4, g ’ (x) < g ’ (1/4) = 0
(f ’ (x) – f (x))e – x < 0
f ’ (x) < f (x) in (0, 1/4).
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