Q3 of 168 Page 231

Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:

f (x) = x3 – 6x2 + 9x + 15

f (x) = x3 – 6x2 + 9x + 15

⇒ f’(x) = 3x2 – 12x + 9


Now, f’(x) = 0


⇒ 3x2 – 12x + 9 = 0


⇒ 3(x-1)(x-3) = 0


⇒ x = 1,3


g’’(x) = 6x – 12 =6(x-2)


Now, f’(1) = 6(1-2)=-6 < 0


and f’(3) = 6(3-2) = 6 > 0


Then, by second derivative test,


⇒ x = 1 is point of local maxima and local maximum of f at x = 1 is


f(1) = 13 – 6 +9 +15 = 19


And,


x = 3 is point of local minima and local minimum value of f at x = 3 is


f(3) = 27 – 54 + 27 +15 = 15


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