Question

for medal: i have f''(x)= 6x and after setting that to 0, i got x=0. how do i find the intervals of concavity? do i plug in 0 to f(x), f'(x), or f''(x)?

Answer 1

concave up if f''(x) > 0, and concave down if f''(x) < 0

Answer 2

ok, how would i test the intervals?

Answer 3

i drew a number line and marked 0 in the middle. how do i know if the graph concaves up or down to the left and right of 0?

Answer 4

pick a value either side of the solution...

and test in the 2nd derivative...

Answer 5

your graph has only 1 point of inflexion.

so the intervals will be to - infinity to zero and zero to infinity

the if the curve is concave up to the left then its concave down on the right...

and vice versa...

so test either side

Answer 6

so can i plug in x=-1 into 6x and x=1 in 6x?

Answer 7

yet... you get f''(-1) = -6 which is < 0 so concave down

f"(1) = 6 which is > 0 so concave up

Answer 8

when finding the coordinates of a local min/max, i plug in the x value into f(x) and not f'(x) right?

Answer 9

well the values of for the local max and min come from f'(x) set it to zero and solve for x, called stationary points

to determine the nature of the stationary points substitute them into the 2nd derivative

f'' > 0 min

f'' = 0 horizontal point of inflexion

f'' < 0 max

hope this helps

Answer 10

thanks