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Question
If 4a + 2b + c = 0, then the equation 3ax2 + 2bx + c = 0 has at least one real root lying in the interval
Options
(0, 1)
(1, 2)
(0, 2)
none of these
Solution
(0, 2)
\[Let\]
\[f\left( x \right) = a x^3 + b x^2 + cx + d . . . . . \left( 1 \right)\]
\[f\left( 0 \right) = d\]
\[f\left( 2 \right) = 8a + 4b + 2c + d\]
\[ = 2\left( 4a + 2b + c \right) + d\]
\[ = d \left( \because \left( 4a + 2b + c \right) = 0 \right)\]
f is continuous in the closed interval [0, 2] and f is derivable in the open interval (0, 2).
Also, f(0) = f(2)
By Rolle's Theorem,
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