Example 1Below is a plot of f(x) = x^3  3x^2  24x + 5 (in red) and f'(x) = 3x^2  6x  24 (in green).The critical points are x = 2 and x = 4. We see that the local extrema occur at critical points. Example 1Below is a plot of f(x) = x^3  3x^2  24x + 5 (in red) and f'(x) = 3x^2  6x  24 (in green).The critical points are x = 2 and x = 4. We see that the local extrema occur at critical points. 
Example 2Below is a plot of g(x) = 3\sqrt[3]{x}(x^2  7) (in red) and f'(x) = 7x^{2/3}(x^2  1) (in green).The critical points are x = 1, x = 1, and x = 0. We see that the local extrema occur at critical points (x = 1, 1), but not every critical point yields a local extremum: there is no local extremum at x = 0. Example 2Below is a plot of g(x) = 3\sqrt[3]{x}(x^2  7) (in red) and f'(x) = 7x^{2/3}(x^2  1) (in green).The critical points are x = 1, x = 1, and x = 0. We see that the local extrema occur at critical points (x = 1, 1), but not every critical point yields a local extremum: there is no local extremum at x = 0. 
Example 3Below is a plot of q(x) = x (in red) and q'(x) = x/x (in green).The only critical point is at x = 0. We see that the local extrema occur at critical points. Example 3Below is a plot of q(x) = x (in red) and q'(x) = x/x (in green).The only critical point is at x = 0. We see that the local extrema occur at critical points. 
Example 4Below is a plot of r(x) = x^3 + 1 (in red) and r'(x) = 3x^2 (in green).The only critical point is at x = 0. We see that there nevertheless is no local extrema at x =0. Example 4Below is a plot of r(x) = x^3 + 1 (in red) and r'(x) = 3x^2 (in green).The only critical point is at x = 0. We see that there nevertheless is no local extrema at x =0. 
Example 5Below is a plot of y = x  \cos x (in red) and \frac{dy}{dx} = 1+\sin x (in green) on [0, 2\pi].We see the minimum value is at c = 0 and the maximum value is at c = 2\pi. Example 5Below is a plot of y = x  \cos x (in red) and \frac{dy}{dx} = 1+\sin x (in green) on [0, 2\pi].We see the minimum value is at c = 0 and the maximum value is at c = 2\pi. 
Example 6Below is a plot of y = x^3 2x  1/2 (in red) and \frac{dy}{dx} = 3x^2  4 (in green) on [1, 2].We see the minimum value is at c = \sqrt{2/3} and the maximum value is at c = 2. Example 6Below is a plot of y = x^3 2x  1/2 (in red) and \frac{dy}{dx} = 3x^2  4 (in green) on [1, 2].We see the minimum value is at c = \sqrt{2/3} and the maximum value is at c = 2. 

