Let f ( x ) be a continuous function which satisfies, f 3 ( x ) − 5 f 2 ( x ) + 10 f ( x ) − 12 ≤ 0 , f 2 ( x ) − 4 f ( x ) + 3 ≥ 0 , f 2 ( x ) − 6 f ( x ) + 8 ≤ 0 . If the equation to the tangent dra

Question

Let f ( x ) be a continuous function which satisfies, f 3 ( x ) − 5 f 2 ( x ) + 10 f ( x ) − 12 ≤ 0 , f 2 ( x ) − 4 f ( x ) + 3 ≥ 0 , f 2 ( x ) − 6 f ( x ) + 8 ≤ 0 . If the equation to the tangent drawn from ( 2 , 0 ) to the curve, y = x 2 f ( sin x ) is of the form, y = αx − 2 α , α ∈ R + , then the numerical value of 6 α ​ is.

TestHub · Accepted Answer

f 2 − 6 f + 8 ≤ 0 ⇒ 2 ≤ f ≤ 4 ; f 2 − 4 f + 3 ≥ 0 ⇒ f ≤ 1 or f ≥ 3 ; so f ∈ [ 3 , 4 ] . Let g ( f ) = f 3 − 5 f 2 + 10 f − 12 ; g ′ ( f ) = 3 f 2 − 10 f + 10 > 0 always, and g ( 3 ) = 0 , so g ( f ) ≤ 0 ⇒ f ≤ 3 . Hence f ( x ) = 3 for all x , and the curve is y = 3 x 2 . Tangent at ( t , 3 t 2 ) : y = 6 t x − 3 t 2 . It passes through ( 2 , 0 ) : 12 t − 3 t 2 = 0 ⇒ t = 4 (since α ∈ R + ) y = 24 x − 48 ⇒ α = 24 ⇒ 6 α ​ = 4

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