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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.

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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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256. Find all points of local maxima and local minima of the function f given by f (x) = x³– 3x + 3.

f (x) = x³– 3 x + 3
∴ f ' (x) – 3 x² 3 = 3 (x – 1) ( v + 1)
f ' (x) = 0 ⇒ 3 x² – 3 = 0 ⇒ x²– 1 – 0 ⇒ x² = 1 ⇒ v = – 1, 1
When x < – 1 slightly, f ' (x) = 3 ( ve) ( ve ) = + ve
When x > – 1 slightly, f ' (x) = 3 ( ve) ( + ve) = ve
∴ at x = – 1, f ' (x) changes from + ve to ve
∴ f (x) has local maximum value at x = 1
When x < 1 slightly, f ' (x) = 3 (– ve) ( + ve) = – ve
When x > 1 slightly, f ' (x) = 3 ( + ve) (+ ve) = + ve
∴ at x = 1, f ' (x) changes from ve to + ve
∴ f (x) has local minimum value at x = 1