Question
If $$\int_0^1 {\left( {1 + {{\sin }^4}x} \right)\left( {a{x^2} + bx + c} \right)dx} = \int_0^2 {\left( {1 + {{\sin }^4}x} \right)\left( {a{x^2} + bx + c} \right)dx} $$ then the quadratic equation $$a{x^2} + bx + c = 0$$ has :
A.
at least one root in (1, 2)
B.
no root in (1, 2)
C.
two equal roots in (1, 2)
D.
both roots imaginary
Answer :
at least one root in (1, 2)
Solution :
Clearly, $$\int_1^2 {\left( {1 + {{\sin }^4}x} \right)} \left( {a{x^2} + bx + c} \right)dx = 0$$
As $$1 + {\sin ^4}x > 0$$ for all $$x\, \in \left[ {1,\,2} \right],$$ the above can be true only if $${a{x^2} + bx + c}$$ changes sign in [1, 2]. So, $$a{x^2} + bx + c = 0$$ has at least one real root between 1 and 2.