Just the sign of f'(x) tells you where the graph is heading
You've learned how to find a derivative — now it's time to see what you can do with it. Use the sign of f'(x) to find where a graph increases or decreases and its extrema, use the sign of f''(x) to find concavity and inflection points, and finish by solving a real optimization problem.
Here, f(x) = x³/3 − x. Move x with the slider and check the sign of the tangent's (red line) slope — that is, of f'(x). If f'(x)>0, the function is increasing; if f'(x)<0, it's decreasing. The point where f'(x)=0 and the sign actually flips is an extremum (a local max or local min). This same principle of finding increasing/decreasing intervals and extrema is used directly in real-world problems too, like finding where profit is maximized or the instant a thrown ball reaches its highest point.
f(x)TangentExtremum
x0.00
f(x) = x³/3 − x · f'(x) = x² − 1
Concavity is determined by the sign of f''(x). If f''(x)>0, the graph is concave up (bowl-shaped); if f''(x)<0, it's concave down (mountain-shaped). The point where the sign flips is an inflection point. If f'(x) tells you the slope of the graph, f''(x) tells you whether that slope is getting bigger or smaller — in other words, which direction the curve is bending.
Current positionConcave upConcave downInflection point
x0.00
f''(x) = 2x
Optimization problem — There's a rectangular garden fenced in with a fixed perimeter of 20m. Change the width w with the slider, and find for yourself the point where the area A(w) is largest. The point where the derivative A'(w)=0 is exactly the answer. This method of finding where a derivative becomes 0 is used directly in optimization problems everywhere in real life — maximizing the volume of a box, minimizing cost, and so on.