Step-by-step explanation: (a) To find the derivative of f(x), we differentiate each term of the function:
f(x) = x³ + 6x² - 36x + 8
Using the power rule, the derivative is:
f'(x) = 3x² + 12x - 36
Therefore, f'(x) = 3x² + 12x - 36.
(b) To find the second derivative of f(x), we differentiate the derivative f'(x):
f'(x) = 3x² + 12x - 36
Differentiating each term:
f''(x) = 6x + 12
Therefore, f''(x) = 6x + 12.
(c) To determine where f is increasing, we need to find where the first derivative is positive (greater than 0).
Setting f'(x) > 0 and solving for x:
3x² + 12x - 36 > 0
Factoring the quadratic equation:
3(x² + 4x - 12) > 0
Now, we can find the interval of increase by analyzing the sign of the quadratic expression:
x² + 4x - 12 > 0
The quadratic equation factors as:
(x + 6)(x - 2) > 0
From this, we can determine the sign of the expression for different intervals:
Interval (-∞, -6): (-)(-) > 0, which is false.
Interval (-6, 2): (-)(+) > 0, which is true.
Interval (2, ∞): (+)(+) > 0, which is true.
Therefore, the interval of increase for f(x) is (-6, 2).
(d) To determine where f is decreasing, we need to find where the first derivative is negative (less than 0).
Setting f'(x) < 0 and solving for x:
3x² + 12x - 36 < 0
Using the same quadratic expression, we can determine the sign of the expression for different intervals:
Interval (-∞, -6): (-)(-) < 0, which is true.
Interval (-6, 2): (-)(+) < 0, which is false.
Interval (2, ∞): (+)(+) < 0, which is false.
Therefore, the interval of decrease for f(x) is (-∞, -6).
(e) To determine where f is concave downward, we need to find where the second derivative is negative (less than 0).
Setting f''(x) < 0 and solving for x:
6x + 12 < 0
Simplifying the inequality:
x + 2 < 0
Solving for x:
x < -2
Therefore, the interval of downward concavity for f(x) is (-∞, -2).
(f) To determine where f is concave upward, we need to find where the second derivative is positive (greater than 0).
Setting f''(x) > 0 and solving for x:
6x + 12 > 0
Simplifying the inequality:
x > -2
Therefore, the interval of upward concavity for f(x) is (-2, ∞).