Question

f(x) = 3x^4 - 4x^3
f'(x) = 12x^3 - 12x^2
f''(x) = 36x^2 - 24x

Can you please teach me how to find increasing decreasing concave up/down infection points?

Answer 1

@rulnick

Answer 2

@Mertsj @zzr0ck3r

Answer 3

First let us factorize the original function and find its roots

Answer 4

can you do that?

Answer 5

As you can see I already did that

Answer 6

the roots for all 3? or just the first and second derivative?

Answer 7

1. If
f '(x) > 0
for all x on (a,b),
then f is
increasing
on [a,b]
2. If
f '(x) < 0
for all x on (a,b),
then f is
decreasing
on [a,b]
3. If
f '(x) = 0
for all x on (a,b),
then f is
constant
on [a,b]

Answer 8

I am a bit confused :S hold on

Answer 9

But how can I determine that info you provided above that easily

Answer 10

You could graph the first derivative. Or you could set it equal to 0 and solve and then test each interval to see if the derivative is positive or negative within the interval.

Answer 11

uhm how about concave up/down - inflection points?

Answer 12

That takes us to the second derivative. If the second derivative is positive, the function is concave upward. Downward if negative. Inflection point if 0

Answer 13

Concavity Theorem:

If the function f is twice differentiable at x=c, then the graph of f is concave upward at (cf(c)) if f(c)0 and concave downward if f(c)0.

Answer 14

arg I think I need more practice, ill try some of these now

Answer 15

Did you understand this? I had to go sorry

Answer 16

factoring helps a bit maybe? or no?

Answer 17

to find whether a function is increasing or decreasing, you need the first derivative.
Graph the first derivative.
For each x-value:
If the first derivative is positive, the function is increasing at that x-value
If the first derivative is negative, the function is decreasing at that x-value
If the first derivative is zero, the function is neither increasing nor decreasing at that x-value.
to find concavity and inflection points, you need the second derivative.
Graph the second derivative.
for each x-value:
If the second derivative is positive, the function is concave up at that x-value
If the second derivative is negative, the function is concave down at that x-value
If the second derivative is zero (and not constant) the function has a point of inflection at that x-value