Question

f(x)=x^3-6x+1

find the critical points, where the function is increasing or decreasing and explain the shape of the graph using the derivative and algebra.

i found the derivative to be 3x^2-6. So the critical points are -1.4 and 1.4. The derivative is decreasing until (0,6) and then it begins increasing. i don't know how to go about explaining the shape of the graph with algebra though. is it related to quadratic equations???

Answer 1

yes. Now, you need the second derivative, which in this cae is 6x. So, at the first "critical point,-1.4, the second derivative is negative, so at that point, so it is a maximum. At the second point 1.4, the second derivative is positive, so the graph is a relative minumum there. So, the graph curves up till it hits -1.4, then it curves down unitl it hits 1.4, then it curves up again.