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7-42.

Without a calculator, find the minimum value of f(x) = 2x3 − l5x2 + 24x + 19 for {x: 0 ≤ x ≤ 5}. Use the second derivative to justify that your value is a minimum. Homework Help ✎

Find f '(x) and f ''(x).

Identify candidates for minimum values by finding where f '(x) = 0 and where f '(x) = DNE. Note: this includes the endpoints.

Evaluate all candidates (except for the endpoints) in the 2nd derivative.
Recall that a LOCAL minimum will exist where f '(candidate) = 0 or DNE and f ''(candidate) > 0.
Note: endpoints can be global (not local) minima.

To find the GLOBAL minimum, evaluate and compare f (local minimum) with f (endpoint) and f (endpoint b).
The lowest value is the global minimum.