A graph that is large enough to show a function’s global behavior may fail to reveal important local features. The graph of ƒ(x) = (x⁸/8) - (x⁶/2) - x⁵ + 5x³ is a case in point.

a. Graph ƒ over the interval -2.5 ≤ x ≤ 2.5. Where does the graph appear to have local extreme values or points of inflection?

b. Now factor ƒ′(x) and show that ƒ has a local maximum at x =³√5 ≈ 1.70998 and local minima at x = ±23 ≈ ±1.73205.

c. Zoom in on the graph to find a viewing window that shows the presence of the extreme values at x = ³√5 and x = √3.

The moral here is that without calculus the existence of two of the three extreme values would probably have gone unnoticed. On any normal graph of the function, the values would lie close enough together to fall within the dimensions of a single pixel on the screen.