Quadratic EquationHard
Question
The greatest value of the function $f(x) = \frac{1}{2bx^{2} - x^{4} - 3b^{2}}$ on the interval $\lbrack - 2,1\rbrack$ depending on the parameter 'b' is/are
Options
A.$- \frac{1}{3b^{2}}$ if $b \in \lbrack 0,2\rbrack$
B.$\frac{1}{4b - 4 - 3b^{2}}$ if $b \in \lbrack 0,4\rbrack$
C.$\frac{1}{8b - 16 - 3b^{2}}$ if $b \leq 2$
D.$- \frac{1}{3b^{2}}$ if $b \geq 2$
Solution
$f(x) = \frac{1}{2bx^{2} - x^{4} - 3b^{2}}\ x \in \lbrack - 2,1\rbrack$
Let $g(t) = 2bt - t^{2} - 3{\text{ }b}^{2}\ t \in \lbrack 0,4\rbrack$
$$g(t) < 0\forall t \in \lbrack 0,4\rbrack$$
Case-I If $b \leq 0$, then $g(t)$ will decrease in $\lbrack 0,4\rbrack$
Maximum value of $f(t) = f(4) = \frac{1}{8\text{ }b - 16 - 3{\text{ }b}^{2}}$
Case
Create a free account to view solution
View Solution FreeMore Quadratic Equation Questions
If both roots of the equation x2 − (m + 1) x + (m + 4) = 0 are negative, then m equals -...The value of maximum real root minus the minimum real root of the equation $\left( x^{2} - 5 \right)^{4} + \left( x^{2} ...If α and β are roots of 2x2 − 3x − 6 = 0, then the equation whose roots are α2 + 2 and β...If α and β are the roots of the equation x2 - x + 1 = 0, then α2009 + β2009 =...Suppose 1, 2, 3 are the roots of the equation $x^{4} + ax^{2} + bx = c$. Then the value of $c$ is...