TTMUA Lab
TMUA 2016 · Paper 1 · Question 18 of 20

TMUA 2016 Paper 1 Question 18

Differentiation and integration — Differentiating a fractional power · where a function decreases. Try it first; the answer and a full worked solution are below.

TMUA 2016 · Paper 1Differentiation and integrationDifferentiating a fractional power · where a function decreases6 options
The function 1−x3√x2 is defined for all x ≠ 0.

The complete set of values of x for which the function is decreasing is

  1. Ax ≤ −2, x > 0
  2. B−2 ≤ x < 0
  3. Cx ≤ 1, x ≠ 0
  4. Dx ≥ 1
  5. E−2 ≤ x ≤ 1, x ≠ 0
  6. Fx ≤ −2, x ≥ 1
Show the answer and worked solution
answer · A
  1. Ax ≤ −2, x > 0
  2. B−2 ≤ x < 0
  3. Cx ≤ 1, x ≠ 0
  4. Dx ≥ 1
  5. E−2 ≤ x ≤ 1, x ≠ 0
  6. Fx ≤ −2, x ≥ 1
Split the fraction into powers before differentiating: f(x) = x−2/3 − x1/3, so f'(x) = −23 x−5/3 − 13 x−2/3 = −x+23x5/3. The sign of x5/3 follows the sign of x, so consider three regions. For x > 0 both x + 2 and x5/3 are positive, so f' < 0 and the function decreases. For −2 < x < 0 the numerator is positive but x5/3 is negative, making f' > 0. For x < −2 both are negative, so f' < 0 again. The complete set is x ≤ −2 together with x > 0.