Question

If f(x) = 4x − x², x ∈ R, then write the value of f(a + 1) −f(a − 1).

Answer 1

Given:
f(x) = 4x − x², x ∈ R
Now,
f(a + 1) = 4(a + 1) $-$ (a + 1)²
= 4a + 4 $-$ (a² + 1 + 2a)
= 4a + 4 $-$ a² $-$ 1 $-$ 2a
= 2a $-$ a² + 3
f(a $-$ 1) = 4(a $-$ 1) $-$ (a $-$ 1)²
= 4a $-$ 4 $-$ (a² + 1 $-$ 2a)
= 4a $-$ 4 $-$ a² $-$ 1 + 2a
= 6a $-$ a² $-$ 5
Thus,
f(a + 1) − f(a − 1) = ( 2a $-$ a² + 3) $-$ (6a $-$ a² $-$ 5)
= 2a $-$ a² + 3 $-$ 6a + a² + 5
= 8 $-$ 4a
= 4(2 $-$ a)