2 A parabola property? ? L? et f? (x? ) = x a.? Show that ? fo?r all? ? y? . b.? Is this prope?rty? true fo?r f? ) = ? ?2 , where a ? is a nonzero real number? c.? Give a geometrical interpretation of this properly. d.? Is this pr?o? erty tru? e for f? (?x? x ?

Solution Step 1: 2 Given function is f(x)=x (a) We have to show that f(x)f(=f (x+y) for all x =/ y xy 2 Here f(x)=x 2 f(y)=y 2 Step 2: d f’(x)= dx(f(x)) = d (2x) dx =2x x+y x+y f ( 2 )=2 2 =x + y …..(1) f(x)f(y) x y2 Now let xy = xy (xy)(x+y) = xy =x+y …..(2) From (1) and (2) f(x)f(y=f (x+y) for all x =/ y xy 2 Step 3: (b) this property true for f ) x2 Here f(x)=ax 2 f(y)=ay 2 Step 4: d f’(x)= dx(f(x)) d 2 = dx(ax ) =2ax x+y x+y f ( 2 )=2a 2 =a(x + y) …..(3) f(x)f(y) ax ay2 Now let xy = xy ax ay2 = 2y2 = a(x y ) xy = a(xy)(x+y) xy =a(x+y) …..(4) From (1) and (2) f(x)f(y=f (x+y ) for all x / y xy 2