Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/193



Examine $$x^3 - 9x^2 + 24x - 7$$ for maximum and minimum values.




 * Solution.
 * style="text-align: center;"|$$f(x) = x^3 - 9x^2 + 24x - 7$$.
 * style="text-align: center;"|$$f'(x) = 3x^2 - 18x + 24$$.
 * Solving
 * style="text-align: center;"|$$3x^2 - 18x + 24 = 0$$
 * colspan="2"|gives the critical values x = 2 and x = 4. &there4; $$f'(2) = 0$$, and $$f'(4) = 0$$.
 * Differentiating again,
 * style="text-align: center;"|$$f''(x) = 6x - 18$$.
 * colspan="2"|Since $$f''(2) = -6$$, we know from (47) that $$f(2) = 13$$ is a maximum.
 * colspan="2"|Since $$f''(4) = +6$$, we know from (48) that $$f(4) = 9$$ is a minimum.
 * }
 * Differentiating again,
 * style="text-align: center;"|$$f''(x) = 6x - 18$$.
 * colspan="2"|Since $$f''(2) = -6$$, we know from (47) that $$f(2) = 13$$ is a maximum.
 * colspan="2"|Since $$f''(4) = +6$$, we know from (48) that $$f(4) = 9$$ is a minimum.
 * }
 * colspan="2"|Since $$f''(4) = +6$$, we know from (48) that $$f(4) = 9$$ is a minimum.
 * }

Examine $$e^x + 2 \cos x + e^{-x}$$ for maximum and minimum values.




 * Solution.
 * style="text-align: center;"|$$f(x) = e^x + 2 \cos x + e^{-x}$$,
 * style="text-align: center;"|$$f'(x) = e^x - 2 \sin x - e^{-x} = 0$$, for x = 0,
 * style="text-align: center;"|$$f(x) = e^x - 2 \cos x + e^{-x} = 0$$, for x'' = 0,
 * style="text-align: center;"|$$f'(x) = e^x + 2 \sin x - e^{-x} = 0$$, for x'' = 0,
 * style="text-align: center;"|$$f^{iv}(x) = e^x + 2 \cos x + e^{-x} = 4$$, for x = 0.
 * }
 * style="text-align: center;"|$$f(x) = e^x - 2 \cos x + e^{-x} = 0$$, for x'' = 0,
 * style="text-align: center;"|$$f'(x) = e^x + 2 \sin x - e^{-x} = 0$$, for x'' = 0,
 * style="text-align: center;"|$$f^{iv}(x) = e^x + 2 \cos x + e^{-x} = 4$$, for x = 0.
 * }
 * style="text-align: center;"|$$f^{iv}(x) = e^x + 2 \cos x + e^{-x} = 4$$, for x = 0.
 * }
 * style="text-align: center;"|$$f^{iv}(x) = e^x + 2 \cos x + e^{-x} = 4$$, for x = 0.
 * }


 * Hence from (48), $$f(0) = 4$$ is a minimum.

EXAMPLES

Examine the following functions for maximum and minimum values, using the method of the last section:

4. Investigate $$x^6 - 5x^4 + 5x^3 - 1$$, at x = 1 and x = 3.

5. Investigate $$x^3 - 3x^2 + 3x + 7$$, at x = 1.

6. Show that if the first derivative of $$f(x)$$ which does not vanish for x = a is of odd order (= n), then $$f(x)$$ is an increasing or decreasing function when x = a, according as $$f^{(n)}(a)$$ is positive or negative.