is f(x) is concave up or down if f ''(x) is less than zero? how do you know?

Answer 1

concave up is a "U" shaped area of the curve where slope is everywhere increasing.

from above you're sort of staring down into a "cave"

concave down is an upside down U shape, where slope is everywhere decreasing

2nd derivative, f"(x), gives the change in slope, it's + , f"(x)>0, if the slope is increasing,

it's -, negative, f"(x)<0 if the slope is decreasing, so it's concave downward

graphically, the shape is like an upside down "U",

from below you're staring up into a "cave"

to determine if a point is in the concave downward area, draw a tangent line to the graph at that point. If it lies all above the graph, it's concave downward. If it lies all below the graph, it's concave upward.

if it cuts the graph, it's an inflection point, the point where the graph changes from concave up to concave down or the reverse. The inflection point is where the 2nd derivative = 0