Final answer:
To find extrema for the function f(x) = x3 - 16x - 5, locate critical points by setting the first derivative to zero, then use the second derivative to classify them as minima or maxima. Global extrema occur at these points or at the interval's endpoints.
Step-by-step explanation:
To find all local and global extrema for the polynomial function f(x) = x3 - 16x - 5, we first need to find the function's critical points by taking its derivative and setting it equal to zero. The derivative of the function is f'(x) = 3x2 - 16. Setting this derivative equal to zero gives us the critical points: f'(x) = 0 leads to 3x2 - 16 = 0.
Solving this equation for x gives us the values x = -√(rac{16}{3}) and x = √(rac{16}{3}). To determine whether these points are local maxima, minima or inflection points, we examine the second derivative f''(x) = 6x. If f''(x) is positive at a critical point, it's a minimum; if negative, it's a maximum. For x < 0, f''(x) is negative indicating a local maximum, and for x > 0, f''(x) is positive indicating a local minimum.
The global extrema can be determined by evaluating the function at the critical points and at the endpoints of the given interval. Since the polynomial is of the third degree and has no bounds on its maximum or minimum values, the global extrema within the restricted interval will occur either at the critical points or at the endpoints of the interval.