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Answer:
The interval over which the graph is decreasing is;
(-∞, -3)
Explanation:
The given function is f(x) = 2·(x + 3)² + 2
By expanding the function, we have;
2·x² + 12·x + 20
From the characteristics of a quadratic equation, we have;
The shape of a quadratic equation = A parabola
The coefficient of x² = +2 (positive), therefore the parabola opens up
The parabola has a minimum point
Points to the left of the minimum point are decreasing
The minimum point is obtained as the x-coordinate value when f'(x) = 0
∴ f'(x) = d(2·x² + 12·x + 20)/dx = 4·x + 12
At the minimum point, f'(x) = 4·x + 12 = 0
∴ x = -12/4 = -3
Therefore;
The graph is decreasing over the interval from -infinity (-∞) to -3 which is (-∞, -3)
Please find attached the graph of the function created with Microsoft Excel
The graph is decreasing over the interval (-∞, -3).
