Question

See the ​attached file.

Answer 1

f(x) = 4x - x^2

1)

f(4) = 4(4) - 4^2 = 16 - 16 = 0

f(-4) = 4(-4) - (-4)^2 = -16 - 16 = -32

f(4) - f(-4) = 0 - (-32) = 32

2)

f(3/2) = 4(3/2) - (3/2)^2

f(3/2) = 6 - 9/4 = 15/4

√f(3/2) = √(15/4) = √15 / 2

3)

f(x + h) = 4(x + h) - (x + h)^2

= 4x + 4h -(x^2 + 2xh + h^2)

= 4x + 4h -x^2 - 2xh - h^2

f(x - h) = 4(x - h) - (x - h)^2

= 4x - 4h -(x^2 - 2xh + h^2)

= 4x - 4h -x^2 + 2xh - h^2

So

[f(x + h) -f(x - h) ] / 2h

= [4x + 4h -x^2 - 2xh - h^2 - ( 4x - 4h -x^2 + 2xh - h^2 )] / 2h

=( 4x + 4h -x^2 - 2xh - h^2 - 4x + 4h + x^2 - 2xh + h^2 ) / 2h

= (8h - 4xh) / 2h

= 2h(4 -2x) / 2h

= 4 - 2x

Answer: [f(x + h) -f(x - h) ] / 2h = 4 - 2x