Question

Identify the local minimum value for the function f(x)= 3x^5 - 5x^3 + 2, using the first derivative test.

Answer 1

\[f'(x)=15x^4-15x^2\]
\[f'(x)=15x^2(x^2-1)\]

Answer 2

so far so good?

Answer 3

critical points at
\[\{-1,0,1\}\]

Answer 4

now
\[x^2\geq0\] so the derivative doesn't change sign there. it changes sign at -1 and 1.

Answer 5

goes from being positive to negative at -1 so function is increasing then decreasing means -1 is a local max.
doesn't change sign at 0 so neither max nor min
goes from being negative to positive at x = 1 means function is decreasing then increasing so local min

Answer 6

in sum
-1 is a local max
0 is neither max nor min
1 is local min
function is increasing on
\[(-\infty,-1)\] decreasing on
\[(-1,1)\] and then increasing on
\[(1,\infty)\]

Answer 7

clear enough?

Answer 8

Crystal! :D