### Home > CALC > Chapter 5 > Lesson 5.1.3 > Problem5-27

5-27.

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

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

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.

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

$(0, 40)$ 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.