(1 point) Find the inflection points of f(x)=4x4+22x3−18x2+10. (Give your answers as a comma separated list, e.g., 3,-2.)

2 answers

(1 point) Is the graph of y=sin(x4) increasing or decreasing when x=14?
(enter increasing, decreasing, or neither).

Is it concave up or concave down?
(enter up, down, or neither).
ok, I'll help on this one too. But I'm sure your text explains this all in detail.
an inflection point occurs (usually) when f"(x) = 0 and f'(x) ≠ 0

the graph is concave up if f"(x) > 0
concave down if f"(x) < 0

for y = 4x^4+22x^3−18x^2+10
y' = 16x^3+66x^2-36x = 2(8x^3+33x^2-18x)
y" = 2(24x^2+66x-18) = 12(4x^2+11x-3) = 12(x+3)(4x-1)
so, there are inflection points at -3, 1/4 since f' ≠ 0 there

for y = sin(x^4)
y' = 4x^3 cos(x^4)
y" = 4x^2(3cos(x^4) - 4x^4 sin(x^4))
y'(14) = 4*14^3 cos(14^4) > 0 so f is increasing
y"(14) = 4*14^2(3cos(14^4) - 4*14^4 sin(14^4)) < 0 so the graph is concave down