Question

find the absolute maximum and minimum values of the function over the indicated interval.

$f(x)=2 x^{2}+3$
(a) $3,6$
(b) $-6,6$

(a) the absolute maximum value is 75 at $x=6$
(use a comma to separate answers as needed)

the absolute minimum value is $\square$ at $x=\square$
(use a comma to separate answers as needed)

Explanation:

Step1: Analyze the function \(f(x)=2x^{2}+3\)

The function \(f(x)=2x^{2}+3\) is a quadratic function. Its derivative \(f^\prime(x) = 4x\). Setting \(f^\prime(x)=0\), we get \(4x = 0\Rightarrow x = 0\). But for the interval \([3,6]\), the critical point \(x = 0\) is not in this interval.

Step2: Evaluate the function at the endpoints of the interval \([3,6]\)

For \(x = 3\):
\(f(3)=2\times3^{2}+3=2\times9 + 3=18 + 3=21\)
For \(x = 6\):
\(f(6)=2\times6^{2}+3=2\times36+3 = 72 + 3=75\)

Answer:

The absolute minimum value is \(21\) at \(x = 3\)