Grade 10 Maths · Trigonometry · Solve product inequality from graphs

Product Inequalities from Graphs Grade 10

Use two graphs to find where f(x) g(x) or f(x) g(x) is positive or negative by comparing their signs on each interval.

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for which values of x is f(x).g(x)≥0. functions grade 12Benit mathematics

For which values of x is f(x).g(x) greater than zero | GRAPH INTERPRETATION |GRADE 12,11,10PushProgramme

For which values of x will x times f at x be greater than zero. Functions grade 12, 11 and 10Benit mathematics

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Question 5.5 · November 2023 · Paper 1 · National (DBE) · 3 marks

The graphs of

f (x) = - \frac{1}{2} (x - 1)^{2} + 8

and

g (x) = \frac{d}{x}

are drawn below. A point of intersection of

f

and

g

is B, the turning point of

f

. The graph

f

has

x

-intercepts at

(- 3; 0)

and

(5; 0)

and a

y

-intercept at C. $Diagram for this question$

5.5 For which values of

x

will

f (x) . g (x) \leq 0

?(3)

Work it out on paper first. Then mark yourself with the official memo, one tick at a time.

More past paper questions: solve product inequality from graphs

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3. Question 7.4.1 · May/June 2025 · Paper 2 · National (DBE) · 2 marks
In the diagram, the graphs of $f (x) = 2 cos x + 1$ and $g (x) = sin 2 x$ are drawn for the interval $x \in [- 18 0^{\circ}; 18 0^{\circ}]$ .
7.4 Use the graphs to determine the values of x, in the interval $x \in [- 18 0^{\circ}; 18 0^{\circ}]$ for which:
7.4.1 $g (x) \cdot f^{'} (x) < 0$ (2)
Show the memo (2 ticks)Memo
1. ✓
  Analyze the conditions for the product to be negativeFor $g (x) \cdot f^{'} (x) < 0$ , the signs of $g (x)$ and $f^{'} (x)$ must be different (+ and - OR - and +).
  $f^{'} (x)$ represents the gradient of $f$ . It is positive when $f$ is increasing ( $- 18 0^{\circ} < x < 0^{\circ}$ ) and negative when $f$ is decreasing ( $0^{\circ} < x < 18 0^{\circ}$ ).
  
  Condition 1: $g (x) > 0$ and $f^{'} (x) < 0$
  $g (x)$ is positive for $(0^{\circ}; 9 0^{\circ}) \cup (- 18 0^{\circ}; - 9 0^{\circ})$ .
  $f^{'} (x)$ is negative for $(0^{\circ}; 18 0^{\circ})$ .
  Intersection: $x \in (0^{\circ}; 9 0^{\circ})$
  
  Condition 2: $g (x) < 0$ and $f^{'} (x) > 0$
  $g (x)$ is negative for $(9 0^{\circ}; 18 0^{\circ}) \cup (- 9 0^{\circ}; 0^{\circ})$ .
  $f^{'} (x)$ is positive for $(- 18 0^{\circ}; 0^{\circ})$ .
  Intersection: $x \in (- 9 0^{\circ}; 0^{\circ})$
  1 mark · Solve product inequality from graphs
2. ✓
  State final union of intervals $x \in (- 9 0^{\circ}; 0^{\circ}) \cup (0^{\circ}; 9 0^{\circ})$
  
  OR
  
  $- 9 0^{\circ} < x < 0^{\circ}$ or $0^{\circ} < x < 9 0^{\circ}$
  1 mark · Solve product inequality from graphs
4. Question 8.5.3 · May/June 2023 · Paper 1 · National (DBE) · 3 marks
Given: $f (x) = x^{3} + 4 x^{2} - 7 x - 10$
8.5 Use your graph to determine the values of $x$ for which:
8.5.3 $f^{'} (x) \cdot f^{''} (x) \leq 0$ (3)
Show the memo (3 ticks)Memo
1. ✓
  $x \leq - 3, 4$ (A) $x \leq - 3, 4$
  1 mark · Solve product inequality from graphs
2. ✓
  $- 1, 33 \leq x \leq 0, 7$ (A $0, 7$ ) $- 1, 33 \leq x \leq 0, 7$
  
  OR $x \in (- \infty; - 3, 4] \cup [- 1, 33; 0, 7]$
  2 marks · Solve product inequality from graphs
5. Question 6.3.1 · November 2023 · Paper 2 · National (DBE) · 2 marks
In the diagram, the graphs of $f (x) = 2 sin 2 x$ and $g (x) = - cos (x + 45°)$ are drawn for the interval $x \in [0°; 180°]$ . $A (105°; - 1)$ lies on $f$ . $Diagram for this question$
6.3 Determine the values of $x$ , in the interval $x \in [0°; 180°]$ , for which:
6.3.1 $f (x) . g (x) > 0$ (2)
Show the memo (2 ticks)Memo
1. ✓
  $x \in (45°; 90°)$ $x \in (45°; 90°)$ OR $45° < x < 90°$
  2 marks · Solve product inequality from graphs
6. Question 7.4.1 · May/June 2022 · Paper 2 · National (DBE) · 2 marks
In the diagram below, the graphs of $f (x) = \frac{1}{2} cos x$ and $g (x) = sin (x - 3 0^{\circ})$ are drawn for the interval $x \in [- 9 0^{\circ}; 24 0^{\circ}]$ . A and B are the $y$ -intercepts of $f$ and $g$ respectively. $Diagram for this question$
7.4 For which values of $x$ , in the interval $x \in [- 9 0^{\circ}; 24 0^{\circ}]$ , will:
7.4.1 $f (x) . g (x) > 0$ (2)
Show the memo (2 ticks)Memo
1. ✓
  First interval $x \in (3 0^{\circ}; 9 0^{\circ})$ OR $3 0^{\circ} < x < 9 0^{\circ}$
  1 mark · Solve product inequality from graphs
2. ✓
  Second interval $\cup (21 0^{\circ}; 24 0^{\circ}]$ OR $21 0^{\circ} < x \leq 24 0^{\circ}$
  1 mark · Solve product inequality from graphs

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