Question

Find the greatest value of the function Find the greatest and the least values of the following functions.
$f\left(x\right)=x^4-6b{x}^2+b^2$ on the interval [-2, 1] depending on the parameter b.

Answer 1

$f\left(x\right)=x^4-6b{x}^2+b^2$

$f^{\prime }\left(x\right)=4x^3-12bx=4x(4x^2-3b)$

$=4x(x-\sqrt{3b})(x+\sqrt{3b})$

$f\left(0\right)=b^2$

$f\left(\sqrt{3b}\right)=9b^2-18b^2+b^2=-8b^2$

$f(-\sqrt{3b})=9b^2-18b^2+b^2=-8b^2$

$f\left(1\right)=b^2-6b$

$f(-2)=b^2-84b+16$

$fb>0$, gretest value $b^2$

If $0<\sqrt{3b}<1$,

$b<\frac{1}{3}\Rightarrow leastvalue\Rightarrow -8b^2$

$If(\sqrt{3b}<2)\Rightarrow leastvalue\Rightarrow -8b^2$

$If(\sqrt{3b}>2)\Rightarrow leastvalue\Rightarrow b^2-24b+16$