Question

Determine the interval(s) at which f(x) is concave up using f''(x) = -x^2+x+6.

Answer 1

I figured it was (-infinity, -2) (3,infinity), but that's not the answer. Other answer choices are:
a) (-2,3)
b) (-infinity, -3) (2, infinity)
c) (2, infinity)
d) (-infinity, -3)

Answer 2

Concave Up = f"(x) > 0

Maybe finding f"(x) = 0 would be beneficial?

Answer 3

-x^2+x+6=0 -> (-x+2)(x+3) = 0
So x= -3 or 2?

Answer 4

Okay,

Answer 5

Ah! It's positive on the line between -3 and 2! So that means (2, infinity) is the point?

Answer 6

How can that be? How about the other side of -3?

Answer 7

Other side of -3 is negative.

Answer 8

Oh wait, that one would be it wouldn't it? Because it's increasing after -3?

Answer 9

Unless it bounces off the x-axis, it had better change sign.

Answer 10

So it's x = -2, 3 instead?

Answer 11

*(-2, 3) instead?

Answer 12

f"(0) > 0

Thus, unless something magical occurs

f"(x) < 0 for x < -3
f"(x) > 0 for -3 < x < 2
f"(x) < 0 for x > 2

It just alternates at the zeros.