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Q.
The equation $3x^2+4ax+b=0$ has at least one root in (0, 1) if
Application of Derivatives
Solution:
Let $f(x) = x^3 + 2ax^2 + bx$ in $[0, 1]$. since $f(x)$ is a polynomial
$\therefore f(x)$ is continuous in $[0, 1]$ and derivable in $(0, 2)$. Also
$f(0) = 0$, $f(1) = 1 + 2 a + b$
$f(0) = f(1)$ gives $0 = 1 + 2a + b$
i.e., $2a + b + 1 = 0$
$\therefore f(x)$ satisfies all the conditions of Rolle’s Theorem if $2a + b + 1 = 0$
$\therefore f'(x)=0$ has at least one root in $(0,1)$
i. e., $3x^2 + 4ax + b = 0$ has at least one root in $(0,1)$
Hence reqd. condition is $2a + b + 1 = 0$