CPM Homework Help : CALC Problem 7-42

Compute without a calculator Without a calculator, find the minimum value of $f(x) = 2x^3 − 15x^2 + 24x + 19$ for { $x$ : $0 ≤ x ≤ 5$ }. Use the second derivative to justify that your value is a minimum.

Step 1:

Find $f^\prime(x)$ and $f^{\prime\prime}(x)$ .

Step 2:

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

Step 3:

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

Step 4:

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.