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Let f (x) = x 3 – 12 x ∴ f ' (x) =3 x 2 – 12 f ' ' (x) = 0 ⇒ 3 x 2 – 12 = 0 ⇒ x 2 – 4 = 0 ⇒ x = 2, – 2 f ' ' (x) = 6 x At x = 2. f ' ' (x) = 12 > 0 ∴ f (x) has local minimum value at x = 2 and this minimum value = (2) 3 – 12 (2) = – 16 At x = – 2, f ' ' (x) = – 12 < 0 ∴ f (x) has local maximum value x = – 2 and this value = (– 2) 3 – 12 (– 2) = – 8 + 24 = 16.

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Application of Derivatives

Long Answer Type

251. Find the local maxima or local minima, if any, of following functions using the first derivative test only. Find also the local maximum and the local minimum values, as the case may be:

straight x cubed left parenthesis straight x minus 1 right parenthesis squared

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252. Find the local maxima or local minima, if any, of following functions using the first derivative test only. Find also the local maximum and the local minimum values, as the case may be:

straight x cubed left parenthesis 2 straight x minus 1 right parenthesis cubed

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253. Find the local maxima or local minima, if any, of following functions using the first derivative test only. Find also the local maximum and the local minimum values, as the case may be:

negative left parenthesis straight x minus 1 right parenthesis cubed space left parenthesis straight x plus 1 right parenthesis squared

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Short Answer Type

254.
Find all the local maxima or minima of the function
f (x) = x³– 12x.

Let f (x) = x³– 12 x
∴ f ' (x) =3 x² – 12
f ' ' (x) = 0 ⇒ 3 x²– 12 = 0 ⇒ x²– 4 = 0 ⇒ x = 2, – 2
f ' ' (x) = 6 x
At x = 2.
f ' ' (x) = 12 > 0
∴ f (x) has local minimum value at x = 2
and this minimum value = (2)³ – 12 (2) = – 16
At x = – 2, f ' ' (x) = – 12 < 0
∴ f (x) has local maximum value x = – 2
and this value = (– 2)³– 12 (– 2) = – 8 + 24 = 16.

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255. 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.

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