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TMUA 2017 · Paper 1 · Question 16 of 20

TMUA 2017 Paper 1 Question 16

Differentiation and integration — Increasing and decreasing functions · combining sign conditions. Try it first; the answer and a full worked solution are below.

TMUA 2017 · Paper 1Differentiation and integrationIncreasing and decreasing functions · combining sign conditions7 options
The functions f and g are given by f(x) = 3x2 + 12x + 4 and g(x) = x3 + 6x2 + 9x − 8.

What is the complete set of values of x for which one of the functions is increasing and the other decreasing?

  1. Ax ≥ −1
  2. Bx ≤ −1
  3. C−3 ≤ x ≤ −2,   x ≥ −1
  4. Dx ≤ −2,   x ≥ −1
  5. Ex ≤ −3,   −2 ≤ x ≤ −1
  6. Fx ≤ −3,   x ≥ −2
  7. G−2 ≤ x ≤ −1
Show the answer and worked solution
answer · E
  1. Ax ≥ −1
  2. Bx ≤ −1
  3. C−3 ≤ x ≤ −2,   x ≥ −1
  4. Dx ≤ −2,   x ≥ −1
  5. Ex ≤ −3,   −2 ≤ x ≤ −1
  6. Fx ≤ −3,   x ≥ −2
  7. G−2 ≤ x ≤ −1
Find where each derivative is positive. f'(x) = 6x + 12, so f increases for x ≥ −2 and decreases for x ≤ −2. g'(x) = 3x2 + 12x + 9 = 3(x+1)(x+3), so g increases for x ≤ −3 and for x ≥ −1, and decreases on −3 ≤ x ≤ −1. Now take the two cases. With f increasing and g decreasing: x ≥ −2 and −3 ≤ x ≤ −1, giving −2 ≤ x ≤ −1. With f decreasing and g increasing: x ≤ −2 together with x ≤ −3 or x ≥ −1, giving x ≤ −3.