To find the interval where the graph of the function f(x) = x³-3x²-9x + 5 is increasing, we need to find the critical points of the function and determine the sign of the derivative in each interval. The critical points are the points where the derivative is zero or undefined.
To find the derivative of the function, we can use the power rule:
f'(x) = 3x² - 6x - 9
Setting f'(x) = 0 and solving for x, we get:
3x² - 6x - 9 = 0
Dividing both sides by 3, we get:
x² - 2x - 3 = 0
Factoring the quadratic equation, we get:
(x - 3)(x + 1) = 0
Therefore, the critical points are x = -1 and x = 3.
We can now test the sign of the derivative in each interval:
Interval (-∞, -1): f'(x) < 0, so the graph is decreasing.
Interval (-1, 3): f'(x) > 0, so the graph is increasing.
Interval (3, ∞): f'(x) < 0, so the graph is decreasing.
Therefore, the interval where the graph of f(x) = x³-3x²-9x + 5 is increasing is (-1, 3).
Jawab:
Penjelasan dengan langkah-langkah:
To find the interval where the graph of the function f(x) = x³-3x²-9x + 5 is increasing, we need to find the critical points of the function and determine the sign of the derivative in each interval. The critical points are the points where the derivative is zero or undefined.
To find the derivative of the function, we can use the power rule:
f'(x) = 3x² - 6x - 9
Setting f'(x) = 0 and solving for x, we get:
3x² - 6x - 9 = 0
Dividing both sides by 3, we get:
x² - 2x - 3 = 0
Factoring the quadratic equation, we get:
(x - 3)(x + 1) = 0
Therefore, the critical points are x = -1 and x = 3.
We can now test the sign of the derivative in each interval:
Interval (-∞, -1): f'(x) < 0, so the graph is decreasing.
Interval (-1, 3): f'(x) > 0, so the graph is increasing.
Interval (3, ∞): f'(x) < 0, so the graph is decreasing.
Therefore, the interval where the graph of f(x) = x³-3x²-9x + 5 is increasing is (-1, 3).