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

[Dramacon]

Homework Statement

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

Homework 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.

The Attempt at a Solution

Isn't a positive gradient an increasing one?

 

[Stimpon]

An increasing gradient means that the gradient itself is increasing.

 

[Dramacon]

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

 

[Dramacon]

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

 

[e to the i pi]

If the gradient of the gradient is positive, then it is an increasing gradient.
Do you know how to check for that?

 

[Deveno]

Homework Statement

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

Homework 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.

The Attempt at a Solution

Isn't a positive gradient an increasing one?

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.

 

[Dramacon]

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