Question

The function f(x)=2 x³-42 x²+270 x+1 has one local minimum and one local maximum. Use a graph of the function to estimate these local extrema.

Answer 1

To estimate the local extrema of the function f(x) = 2x³ - 42x² + 270x + 1, we can analyze the behavior of the graph. The local maxima occur where the graph changes from increasing to decreasing, and the local minima occur where the graph changes from decreasing to increasing. We can find these points by looking for where the first derivative of the function is equal to zero.

Step-by-step explanation:

To estimate the local extrema of the function f(x) = 2x³ - 42x² + 270x + 1, we can analyze the behavior of the graph. The local maxima occur where the graph changes from increasing to decreasing, and the local minima occur where the graph changes from decreasing to increasing. We can find these points by looking for where the first derivative of the function is equal to zero.

First, we find the first derivative by taking the derivative of f(x) with respect to x: f'(x) = 6x² - 84x + 270. Next, we set this derivative equal to zero and solve for x: 6x² - 84x + 270 = 0.

By using the quadratic formula, we find that the solutions for x are approximately x = 5 and x = 9. Therefore, the function f(x) has a local minimum at x = 5 and a local maximum at x = 9.