Let f(x) = x^5 + 3x^2 − 5x − 7, find the interval(s) where f is increasing and decreasing.

Answer 1

f is increasing where f' > 0

f' = 5x^4+6x-5
f' = 0 at x = -1.2588 and 0.6677
f' changes sign at those roots, so
f' > 0 on (-∞,-1.2588)U(0.6677,∞)
f' < 0 on (-1.2588,0.6677)

a look at the graph shows that f is in/de-creasing on these intervals