Final answer:
To find the local minimum of the function, we need to analyze the critical points and the concavity. By taking the derivative of the function and finding the critical points, we can determine the local minimum and maximum of the function. Substituting x = 7, we get f''(7) = 12(7) - 72 = 72, indicating that x = 7 is a local minimum.
Step-by-step explanation:
The local minimum of the function f(x) = 2x³ -36x² +210x+4 can be found by analyzing the critical points of the function. To find the critical points, we first take the derivative of the function and set it equal to zero. Taking the derivative of f(x) gives us f'(x) = 6x² - 72x +210. Setting f'(x) = 0, we solve the quadratic equation to find the critical points.
By factoring or using the quadratic formula, we find the critical points to be x = 3 and x = 7.
To determine whether these critical points are local minimum or local maximum, we can analyze the concavity of the function at these points. By taking the second derivative, f''(x) = 12x - 72, and evaluating it at x = 3 and x = 7, we can determine the concavity.
Substituting x = 3, we get f''(3) = 12(3) - 72 = -24, indicating that x = 3 is a local maximum. Substituting x = 7, we get f''(7) = 12(7) - 72 = 72, indicating that x = 7 is a local minimum.