Calculus BC: Applications of the Derivative

We first calculate the derivatives of the function:

We see that f'(x) = 0 when x = - 1 , 0 , or 1 , so these are the three critical points of f . We calculate the second derivatives at these points:

so by the second derivative test, f has local minima at -1 and 1 and a local maximum at 0 . Substituting back into the original function yields

so f attains its global minimum of -1 at x = ±1 . It is clear from the graph of f that it has no global maximum. To find the points of inflection, we solve f''(x) = 0 , or 12x ² - 4 = 0 , which has solutions x = ±1/3) ±0.58 . Referring once again to the graph of f , we can check that the concavity does indeed change at these x -values.

We compute the derivatives:

Solving f''(x) = 0 for x , we get the inflection points x = ±1/ 0.71 . This is reasonable when one considers the graph of f .

We need f''(1) = f''(2) = 0 , so we might as well let

We search for a function with this derivative, ultimately obtaining

Similarly, a function with this derivative is given by