Matrices and Determinants 2 Question 32

36. 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=5/2, 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).

(1998, 3M)

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Answer:

Correct Answer: 36.

a=14,b=54

f(x)=14x254x+2

Solution:

  1. 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|

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=5/2.

f(5/2)=05a+b=0(i) Also, f(0)=2c=2 and f(1)=1a+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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