If f(x+y)= f(x) f(y) for all x,y belonging to R and f(1)=2 then, area enclosed by 3|x| + 2|y| <= 8
is :-
a) f(4) sq unit
b) (1/2) f(6) sq unit
c) 1/3 f(6) sq unit
d) 1/3 f(5) sq unit
If f(x+y)= f(x) f(y) for all x,y belonging to R and f(1)=2 then, area enclosed by 3|x| + 2|y| <= 8
is :-
a) f(4) sq unit
b) (1/2) f(6) sq unit
c) 1/3 f(6) sq unit
d) 1/3 f(5) sq unit
f(x+y) = f(x)f(y) is the property of exponential function, so let f(x)=a^(x).
Now, given that f(1)=2, a^(1)=2 ,therefore a=2
So, f(x) = 2^(x). And f(6)=2^(6). So, f(6)/3=2^(6)/3
Now. 3|x|+2|y|≤8 is symmetric about x and y axes because of the modulus on x and y.
So let's find the area in the first quadrant and multiply it by four.
For first quadrant x,y>0, so 3x+2y=8, this is a straight line with x intercept 8/3 and y intercept 4, so the area of triangle made by the coordinate axes and this line is the required area in the first quadrant, and it is equal to (8/3)(4)/2. Now the total area is the area in first quadrant times 4.
Area= 4×8×4/6 =2^(6)/3 = f(6)/3.
f(x+y) = f(x)f(y) is the property of exponential function, so let f(x)=a^(x).
Now, given that f(1)=2, a^(1)=2 ,therefore a=2
So, f(x) = 2^(x). And f(6)=2^(6). So, f(6)/3=2^(6)/3
Now. 3|x|+2|y|≤8 is symmetric about x and y axes because of the modulus on x and y.
So let's find the area in the first quadrant and multiply it by four.
For first quadrant x,y>0, so 3x+2y=8, this is a straight line with x intercept 8/3 and y intercept 4, so the area of triangle made by the coordinate axes and this line is the required area in the first quadrant, and it is equal to (8/3)(4)/2. Now the total area is the area in first quadrant times 4.
Area= 4×8×4/6 =2^(6)/3 = f(6)/3.
