Check monotonocity at following points for [16JM120168]
(i) f(x) = x 3 – 3x + 1 at x = – 1, 2
(ii) f(x) = | x – 1 | + 2 | x – 3 | – | x + 2 | at x = – 2, 0, 3, 5
(iii) f(x) = x 1/3 at x = 0
(iv) f(x) = x 2 + 
at x = 1, 2
(v) f(x) = 
at x = 0
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
f'(x) = 3x 2 – 3
= 3(x – 1) (x + 1)
at x = 1 point of minima ∴ Neither increasing nor decreasing
at x = 2 increasing
x = 1 ∴
x = 2
(ii) at x = – 2 decreasing
at x = 0 decreasing
at x = 3 neither increasing nor decreasing
at x = 5 increasing
(ii) x = – 2
x = 0
at x = 3
at x = 5
(iii) f ′ (x) = 
⇒ f ′ (x) > 0 ∀ x ∈ R – {0}
⇒ strictly increasing at x = 0
(iv) f ′ (x) = 
⇒ strictly increasing at x = 2, neither I nor D at x = 1
x = 2 x = 1
(v) f ′ (x) = 
⇒ f ′ (0 – ) = +5
and f ′ (0 + ) = 3
⇒ strictly increasing at x = 0
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