SOLUTION: Find the (least) number a such that the function f(x) = 2x^3−63x^2+624x+5 is concave up for all x > a. I am so confused! I was going to start by differentiating, but I'

Algebra ->  Functions -> SOLUTION: Find the (least) number a such that the function f(x) = 2x^3−63x^2+624x+5 is concave up for all x > a. I am so confused! I was going to start by differentiating, but I'      Log On


   

Question 994559: Find the (least) number a such that the function
f(x) = 2x^3−63x^2+624x+5
is concave up for all x > a.
I am so confused! I was going to start by differentiating, but I'm not sure that is correct. Is this related to finding area of curves?
I am completely lost..
THANK YOU
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Yes, differentiate.
The function is concave up if the derivative is increasing.
df%2Fdx=6x%5E2-126x%2B624
You can put this in vertex form.
df%2Fdx=6%28x%5E2-21x%29%2B624
df%2Fdx=6%28x%5E2-21x%2B%2821%2F2%29%5E2%29%2B624-6%2821%2F2%29%5E2
df%2Fdx=6%28x-21%2F2%29%5E2%2B2496%2F4-2646%2F4
df%2Fdx=6%28x-21%2F2%29%5E2-150%2F4
df%2Fdx=6%28x-21%2F2%29%5E2-75%2F2
So the vertex occurs at (21%2F2,-75%2F2).
The parabola opens upwards since the quadratic multiplier 6%3E0 and the value at the vertex is a minimum.
So for x%3C21%2F2, the derivative is decreasing, the function is concave down.
For x%3E21%2F2, the derivative is increasing, the function is concave up.
.
.
.