Answer:
The first derivative of f(x) is f’(x) = 6x ^ 2 - 48x + 72.
Setting f’(x) equal to zero, we get 6x ^ 2 - 48x + 72 = 0. Factoring, we get (3x - 12)(2x - 6) = 0. Solving for x, we get x = 4 or x = 2. These are the critical numbers of f(x).
Using the First Derivative Test, we can check the sign of f’(x) on either side of the critical numbers. For example, if we plug in x = 1, we get f’(1) = -30, which is negative. This means that f(x) is decreasing on the interval (0, 2). Similarly, if we plug in x = 3, we get f’(3) = -18, which is also negative. This means that f(x) is still decreasing on the interval (2, 4). However, if we plug in x = 5, we get f’(5) = 42, which is positive. This means that f(x) is increasing on the interval (4, ∞). Therefore, we can conclude that x = 4 is a local minimum of f(x), and x = 2 is neither a local maximum nor a local minimum.
You can also use a graphing calculator4 or a website5 to plot the function and see its shape. Here is a graph of f(x):
graph = 2x ^ 3 - 24x ^ 2 + 72x + 6"))
You can see that there is a local minimum at x = 4 and no local maximum.
To answer your question, this function has a local minimum at x = 4, with output value: f(4) = -78, and no local maximum.
I hope this helps you understand how to find the local extrema of a function.
Step-by-step explanation:The first derivative of f(x) is f’(x) = 6x ^ 2 - 48x + 72.
Setting f’(x) equal to zero, we get 6x ^ 2 - 48x + 72 = 0. Factoring, we get (3x - 12)(2x - 6) = 0. Solving for x, we get x = 4 or x = 2. These are the critical numbers of f(x).
Using the First Derivative Test, we can check the sign of f’(x) on either side of the critical numbers. For example, if we plug in x = 1, we get f’(1) = -30, which is negative. This means that f(x) is decreasing on the interval (0, 2). Similarly, if we plug in x = 3, we get f’(3) = -18, which is also negative. This means that f(x) is still decreasing on the interval (2, 4). However, if we plug in x = 5, we get f’(5) = 42, which is positive. This means that f(x) is increasing on the interval (4, ∞). Therefore, we can conclude that x = 4 is a local minimum of f(x), and x = 2 is neither a local maximum nor a local minimum.
You can also use a graphing calculator4 or a website5 to plot the function and see its shape. Here is a graph of f(x):
graph = 2x ^ 3 - 24x ^ 2 + 72x + 6"))
You can see that there is a local minimum at x = 4 and no local maximum.
To answer your question, this function has a local minimum at x = 4, with output value: f(4) = -78, and no local maximum.
I hope this helps you understand how to find the local extrema of a function.