# Inflection Points question

1. Nov 12, 2009

### Slimsta

1. The problem statement, all variables and given/known data
f(x)=-8x4-5x3 +3 for concavity and inflection points.

2. Relevant equations
f(x)=-8x4-5x3 +3
f'(x)=-32x3-15x2
f''(x)=-96x2-30x

3. The attempt at a solution
can someone explain to me what exactly Inflection Points are?
is it like critical points for the first derivative but those are for the second one?
i mean, making f''(x) = 0 and get 2 numbers and those are the Inflection Points?

2. Nov 12, 2009

### LumenPlacidum

Indeed. Inflection points are where the function changes from being concave-up to concave-down or vice versa. This is the same as the local extrema of the derivative, or the roots of the second derivative.

3. Nov 12, 2009

### Slimsta

so i would do this
f''(x)=-96x2-30x
f''(x)=-3x(32x-10)

x=0, 10/32

and those are my answers? well i tried and it doesnt work.. :(

4. Nov 12, 2009

### Staff: Mentor

f''(x) = -96x2 - 30x = -6x(16x + 5)
So f''(x) = 0 for x = 0 and x = -5/16

5. Nov 12, 2009

### LCKurtz

No, it isn't quite the same. The roots of the second derivative are the values of x where there may but there need not be an inflection point. You must always additionally check that the second derivative actually changes sign at those roots. For example, if

$$f''(x) = (x-2)(x-5)^2$$

then x = 2 gives location for an inflection point but x = 5 does not.

6. Nov 12, 2009

### Slimsta

oh yeah i didnt notice the - sign mistake :S

there is a second part to this question which asks to find the points where it concaves up and down.
the graph of f is only concaving up isnt it?

7. Nov 12, 2009

### Staff: Mentor

The graph of f is concave up (concaving isn't a word) when f''(x) > 0, and is concave down when f''(x) < 0. For your problem, f''(x) is positive for some x values and negative for other x values.

8. Nov 12, 2009

### Slimsta

this is how the question is constructed:
"f is concave down on (-infty ,___)U(___, infty) and its concave up on (___,___)"

since f''(x) = 0 for x = 0 and x = -5/16
so when you said "f''(x) is positive for some x values and negative for other x values"
==> x = 0, -5/16

so would it be:
"f is concave down on (-infty ,-5/16)U(0, infty) and its concave up on (-5/16,0)"
?

9. Nov 12, 2009

### Staff: Mentor

Works for me.

10. Nov 12, 2009