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Question: The smaller area enclosed by y = f(x), when f(x) is polynomial of least degree satisfying <img src="...

The smaller area enclosed by y = f(x), when f(x) is polynomial of least degree satisfying = e and the circle x2 + y2= 2 above the x axis is

A

π2\frac { \pi } { 2 }

B

35\frac { 3 } { 5 }

C

π2−35\frac { \pi } { 2 } - \frac { 3 } { 5 }

D

π2+35\frac { \pi } { 2 } + \frac { 3 } { 5 }

Answer

π2−35\frac { \pi } { 2 } - \frac { 3 } { 5 }

Explanation

Solution

lim⁡x→0[1+f(x)x3]1/x\lim _ { x \rightarrow 0 } \left[ 1 + \frac { f ( x ) } { x ^ { 3 } } \right] ^ { 1 / x } exists so lim⁡x→0f(x)x3\lim _ { x \rightarrow 0 } \frac { f ( x ) } { x ^ { 3 } } = 0 It means f(x) =

a4x4 + a5 x5 +……..+ anxn , an ¹ 0 n ³ 4

f(x) is of least degree f(x) = a4x4

lim⁡x→0\lim _ { x \rightarrow 0 }= e, a4 = 1, f(x) = x4 The graph of y = x4 and x2 + y2 = 2

Area = 2 dx

= π2\frac { \pi } { 2 } – 35\frac { 3 } { 5 }