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- Christos

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?

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- Christos

- katieb

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- Christos

- rajee_sam

First let us factorize the original function and find its roots

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- rajee_sam

can you do that?

- Christos

As you can see I already did that

- Christos

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

- 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]

- Christos

I am a bit confused :S hold on

- Christos

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

- 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.

- Christos

uhm how about concave up/down - inflection points?

- 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

- 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.

- Christos

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

- rajee_sam

Did you understand this? I had to go sorry

- Christos

factoring helps a bit maybe? or no?

- anonymous

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

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