Find a continuous function f(x) satisfying x 4 –4x 2 ≤ f(x) ≤ 2x 2 –x 3 for all x ∈ [0, 2] such that the area bounded by y = f(x), y= x 4 –4x 2 , the y- axis and the line x = t (0 ≤ t ≤ 2) is k times the area bounded by y = f(x), y = 2x 2 –x 3 , y- axis and the line x = t(0 ≤ t ≤ 2)
Text Solution
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Sol. Let A 1 be the area bounded by y = f(x), y = x 4 –4x 2 , y- axis and x = t.
Since f(x) ≥ x 4 –2x 2 for all x ∈ [0, 2]
∴ A 1 = 
... (i)
Let A 2 be the area bounded by y = f(x), y = 2x 2 –x 3 , y- axis and x = t. Then,
A 2 = 
It is given that A 1 = kA 2
⇒ 
Differentiating both sides w.r.t. t, we get
f(t) – (t 4 – 4t 2 ) = k [(2t 2 –t 3 ) – f(t)]
⇒ (1+ k) f(t) = k (2t 2 – t 3 ) + (t 4 – 4t 2 )
⇒ f(t) = 
(2t 2 –t 3 ) + 
for all t ∈ [0, 2]
Hence f(x) = 
(2x 2 – x 3 ) + 
= 
{x 4 – kx 3 + 2(k – 2) x 2 }
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