If f ′′ (x) > 0 ∀ x ∈ (a, b), then the curve y = f(x) is concave up (or convex down) in (a,b) and
If f ′′ (x) < 0 ∀ x ∈ (a, b) then the curve y = f(x) is concave down (or convex up) in (a, b).
Inflection point :
The point where concavity of the curve changes is known as point of inflection (at inflection
point f ′′ (x) is equal to 0 or undefined).
(i) Number of point of inflection for f(x) = (x – 1) 3 (x– 2) 2 , is
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
(i) y = (x – 1) 3 (x – 2) 2
= 3(x – 1) 2 (x – 2) 2 + 2(x – 2) (x – 1) 3
= (x – 1) 2 (x – 2) [3(x – 2) + 2(x – 1)]
= (x – 1) 2 (x – 2) (5x – 8)
(x 2 – 2x + 1) (5x 2 – 18x + 16)
= (2x – 2) (5x 2 – 18x + 16) + (10x – 18) (x 2 – 2x + 1) = 0
= 20x 3 – 42x 2 + 11x – 50 = 0
= 10x 3 – 42x 2 + 57x – 25 = 0
(x – 1) (10x 2 – 32x + 25) = 0
x = 1 or x = 
no. of points of inflections = 3
(ii) f(x) = x 4 + ax 3 + 
f ′ (x) = 4x 3 + 3ax 2 + 3x
f ′′ (x) = 12x 2 + 6ax + 3
Now f(x) will be concave upward along the entire
real line iff f ′′ (x) ≥ 0 ∀ x ∈ R
12x 2 + 6ax + 3 > 0 ⇒ D ≤ 0
36a 2 – 144 ≤ 0
a 2 – 4 ≤ 0 ⇒ a ∈ [– 2, 2]
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