Quadratic EquationHard
Question
The greatest value of the function $f(x) = x^{4} - 6bx^{2} + b^{2}$ on the interval $\lbrack - 2,1\rbrack$ depending on the parameter $b$ is
Options
A.$b^{2}$ if $b \geq \frac{2}{3}$
B.$16 - 24\text{ }b + b^{2}$ if $b \leq \frac{2}{3}$
C.$4 - 12b + b^{2}$ if $0 \leq b \leq \frac{4}{3}$
D.$16 - 24\text{ }b + b^{2}$ if $b \geq \frac{2}{3}$
Solution
$f(t) = t^{2} - 6bt + b^{2},\ t \in \lbrack 0,4\rbrack$
maximum value of $f(t)$
$$= \left\{ \begin{matrix} f(4) = 16 - 24b + b^{2}, & 3b \leq 2 \\ f(\rho) = b^{2}, & b \geq 2 \end{matrix} \right.\ $$
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