Determine where the function is concave upward and where it is concave downward. (Enter your answers using interval notation.) f(x) = 3x4 − 30x3 + x − 5 concave upward concave downward

Answer 1

f^'(x) = 4x³ - 90x² + 1

f^"(x) = 12x² - 180x = 0

12x(x - 15) = 0

x = 0 or x = 15

These will both be change of sign zeros since they are of odd multiplicity (single roots) and since f^" a concave up parabola f^" is + on (-∞ , 0) U (15 , ∞) (and f(x) is concave up there), and f^"(x) is - on (0 , 15) (and f concave down).