Sponsor Area

Application Of Derivatives

Question
CBSEENMA12035451

Find all the points of local maxima and local minima of the function f given by f (x) = x3 – 27 x + 3. Also, find local maximum and local minimum value of f.

Solution

Here f (x) = x3 – 27 x + 3
∴    f ' (x) = 3 x2 – 27
and f ' ' (x) = 6 x
Now f ' (x) = 0 ⇒ 3 x– 27 = 0 ⇒ x2 – 9 = 0
⇒ x2 = 9 x = – 3, 3
At x = 3, f ' ' (x) = 18 > 0
∴  f has a local minima at x = 3
and local minimum value = (3)– 27 (3) + 3 = 27 – 81 + 3 = – 51
At x = – 3, f ' ' (x) = – 18 < 0
∴  f has a local maxima at x = – 3
and local maximum value = (– 3)3 –21 ( 3) + 3 = 27 + 81 + 3 = 57

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