Definite IntegrationHard
Question
Let a, b, c be non-zero real numbers such that ;
(1 + cos8x)(ax2 + bx + c) dx = 
(1 + cos8x)(ax2 + bx + c) dx, then the quadratic equation ax2 + bx + c = 0 has -
(1 + cos8x)(ax2 + bx + c) dx = (1 + cos8x)(ax2 + bx + c) dx, then the quadratic equation ax2 + bx + c = 0 has -Options
A.no root in (0, 2)
B.at least one root in (0, 2)
C.a double root in (0, 2)
D.none
Solution
(1+ cos8x)(ax2 + bx + c) dx=
(1+ cos8x)(ax2 + bx + c) dx +
(1 + cos8x)(ax2 + bx + c) dx∴
(1 + cos8x)(ax2 + bx + c) dx = 0since 1 + cos8 x is always positive
=
f(x)dx = 0 (b > a)means f(x) is positive in some portion and negative
in some portion from a to b
∴ ax2 + bx + c is positive and negative in (1, 2)
∴ ax2 + bx + c has a root in (1, 2)
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