CPM Homework Help : CALC Problem 5-27

Identify the maxima and minima of $f ( x ) = - \frac { 1 } { 4 } x ^ { 4 } + 3 x ^ { 3 } - 10 x ^ { 2 } + 40$ .

Step 1:

Find CANDIDATES for maxima and minima by setting the first derivative equal to $0$ .

Step 2:

Decide if each candidate is a maximum, minimum or neither. You could check for slope change by evaluating the 1st-derivative at points close to each candidate. You could evaluate each candidate in the 2nd-derivative. If the graph is concave up, it cannot have a local max. And vice versa.

Step 3:

Recall that maxima and minima are $y$ -values, not $x$ -values. So use $f(x)$ to find the $y$ -value of each candidate.

Answer:

$(0, 40)$ is a local maximum, $(4, 8)$ is a local minimum, $(5, 8.75)$ is a local maximum, and $(5, 8.75)$ is the global maximum. There is no global minimum.