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Question

Suppose, f(x) is a function satisfying the following conditions
(a) f(0) = 2, f(1) = 1
(b) f has a minimum value at x=52, and
(c) for all x,
f(x)=∣ ∣2ax2ax12ax+b+1bb+112(ax+b)2ax+2b+12ax+b∣ ∣
where a, b are some constants. Determine the constants a, b and the function f(x).
  1. f(x)=14x254x+2
  2. f(x)=x3+7516x27516x+2
  3. f(x)=x
  4. f(x)=14x254x+2


Solution

The correct option is A f(x)=14x254x+2
Given, f(x)=∣ ∣2ax2ax12ax+b+1bb+112(ax+b)2ax+2b+12ax+b∣ ∣
Applying R3R3R12R2, we get
f(x)=∣ ∣2ax2ax12ax+b+1bb+11001∣ ∣=2ax2ax1bb+1=2ax1b1[C2C2C1]     f(x)=2ax+b
On integrating, we get f(x) = ax2 + bx + c,
where c is an arbitrary constant.
Since, f has maximum at x=52.
  f(52)=0    5a+b=0      . . . (i)
Also,  f(0) = 2 c = 2 and f(1) = 1
a + b + c = 1                    . . . (ii)
On solving Eqs. (i) and (ii) for a, b, we get
a=14, b=54
Thus,      f(x)=14x254x+2

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