The question says : the gradient of a curve is give by dy/dx = x^2 - 6x. Find the set of values of x for which y is an increasing function of x.

Question

I didn't understand what they are asking me to do could you solve and explain?

Answer 1

You want x^ - 6x to be positive, so
x^2 - 6x > 0
x(x-6) > 0

so the critical values of x are 0 and 6
look at a number < 0 , say x = -1
your statement is -1(-7) > 0 , which is TRUE
look at a number between 0 and 6, say x=4
your statement is 4(-2) > 0 , which is FALSE
look at a number > 6, say x = 10
your statement is 10(4) > 0 , which is TRUE

so x must be less than 0 OR greater than 6
your function increases for
x < 0 OR x > 6