Let f(x) = 2x^{3}+9. Find the open intervals on which f is increasing (decreasing). Then determine the x-coordinates of all relative maxima (minima).
1. f is increasing on the intervals:
2. f is decreasing on the intervals:
3. The relative maxima of f occur at x =
4. The relative minima of f occur at x =
f' = 6x^2+9
clearly, f' is always positive, so there are no extrema, and no intervals where f is decreasing.
Just think of what you know of the curve y=x^3
To find the open intervals on which f(x) = 2x^3 + 9 is increasing or decreasing, we need to determine where its derivative is positive or negative.
1. To find the intervals where f(x) is increasing, we need to find where its derivative, denoted as f'(x), is positive. Calculate the derivative of f(x) as follows:
f'(x) = d/dx(2x^3 + 9)
= 6x^2
Now, solve the inequality f'(x) > 0 to find where f(x) is increasing:
6x^2 > 0
Since the leading coefficient is positive, the parabola opens upwards, and the function is increasing when x^2 > 0. This means that the function is increasing for all real values of x. Therefore, the answer to 1 is: f is increasing on all real intervals.
2. To find the intervals where f(x) is decreasing, we need to find where its derivative, f'(x), is negative. Solve the inequality f'(x) < 0 as follows:
6x^2 < 0
Since the leading coefficient is positive, the parabola opens upwards, and the function is decreasing only when x^2 < 0. However, there are no real solutions to this inequality. Therefore, the answer to 2 is: f is decreasing on no open intervals.
3. To determine the x-coordinates of any relative maxima, we need to find the critical points of f(x). Critical points occur where the derivative f'(x) is either zero or undefined. In this case, f'(x) is defined for all real values of x. To check if there are any critical points, solve the equation f'(x) = 0:
6x^2 = 0
x^2 = 0
x = 0
Therefore, the only critical point is x = 0. Since f(x) is increasing on all intervals, there are no relative maxima, and the answer to 3 is: There are no relative maxima.
4. Similar to finding the relative maxima, the solution for finding relative minima is the same. However, since there are no relative maxima, there are also no relative minima. Therefore, the answer to 4 is: There are no relative minima.
In summary:
1. f is increasing on all real intervals.
2. f is decreasing on no open intervals.
3. The relative maxima of f occur at x = There are no relative maxima.
4. The relative minima of f occur at x = There are no relative minima.