# What is the difference between an 'increasing gradient' and a positive gradient?

1. Nov 6, 2011

### Dramacon

1. The problem statement, all variables and given/known data
f(x)= 3+6x-2x^3

(a) Determine the values of x for which the graph of f has positive gradient
(b) Find the values of x for which the graph of f has increasing gradient
2. Relevant equations
I had originally thought the two terms meant the same thing, but when I checked the answers at the back of the book, they gave two different answers.

3. The attempt at a solution
Isn't a positive gradient an increasing one?

2. Nov 6, 2011

3. Nov 6, 2011

### Dramacon

So when a line is at more than 45 degrees, you mean?

4. Nov 6, 2011

### Dramacon

Or do you mean increasing from one stage to the next?

5. Nov 6, 2011

### e to the i pi

Do you know how to check for that?

6. Nov 6, 2011

### Deveno

suppose our function was g(x) = 2x + 3.

at any given point, the gradient (slope of the graph) is constant, it is 2.

note that g'(x) = 2 is positive, but it ISN'T increasing, it's flat.

to see whether or not the gradient is increasing/decreasing/neither, you need to find the gradient of the gradient.

in terms of derivatives, this means you need to look at the second derivative, to tell whether the first derivative is increasing, decreasing, or "flat". note that these are "local" properties, the answers you get depend on "which x" you look at.

7. Nov 7, 2011

### Dramacon

Ah, I see! :) Thank you! This makes so much more sense now.