SOLUTION: Determine where the graph of the function is concave up and concave down (if you need to enter use infinity or - infinity) f(x)=x^3-15x^2+12x+6 concave up concave down can so

Question 201601: Determine where the graph of the function is concave up and concave down (if you need to enter use infinity or - infinity)
f(x)=x^3-15x^2+12x+6
concave up
concave down
can someone help please
thanks

Found 2 solutions by jsmallt9, RAY100:
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
Concavity can be found using the second derivative.
f(x)=x^3-15x^2+12x+6
f'(x) = 3x^2 -30x + 12
f''(x) = 6x -30

To find where the graph is concave up we want to find where the second derivative is positive. In other "words", solve
6x - 30 > 0
Adding 30 and then dividing by 6 we get
x > 5
This tells us that the graph is concave up for all x-values that are greater than 5.

Similarly we can find that the graph is concave down for all x-values that are less than 5.
Answer by RAY100(1637) (Show Source): You can put this solution on YOUR website!
y= x^3 -15x^2 +12x +6
.
y' = 3x^2 -30x +12
.
to find vertex, set slope(y') =0
.
3x^2 -30x +12 =0
3(x^2 -10x +4) =0
.
solve using quadratic eqn,,,x= 9.58,,,.417
.
to find y,,,substitute in original eqn,,,,(9.58,-376.5),,,(.417,,8.47)
.
since this is a positive cubic,,rises to right,,
1st vertex,,,(.417,8.47) is concave down,,,and
second vertex (9.58, -376.5) is concave up
.
checking,,,(.5,8.375) (.3,8.277),,,ok cc down
(9.5,-376.4),(9.7, -376.3) ,,,,ok cc up
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