## Christos 2 years ago 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?

1. Christos

@rulnick

2. Christos

@Mertsj @zzr0ck3r

3. rajee_sam

First let us factorize the original function and find its roots

4. rajee_sam

can you do that?

5. Christos

As you can see I already did that

6. Christos

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

7. Mertsj

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]

8. Christos

I am a bit confused :S hold on

9. Christos

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

10. Mertsj

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.

11. Christos

uhm how about concave up/down - inflection points?

12. Mertsj

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

13. Mertsj

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.

14. Christos

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

15. rajee_sam

Did you understand this? I had to go sorry

16. Christos

factoring helps a bit maybe? or no?

17. Peter14

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