Question

find the absolute maximum and minimum values of the function over the indicated interval.
f(x)=2x² + 3
(a)3,6 (b)-6,6
(a) the absolute maximum value is □ at x=□
(use a comma to separate answers as needed.)

Explanation:

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

The function \(y = 2x^{2}+3\) is a parabola. The general form of a parabola is \(y = ax^{2}+bx + c\), here \(a = 2\), \(b = 0\), \(c = 3\). Since \(a=2>0\), the parabola opens upwards. The vertex of the parabola \(y = ax^{2}+bx + c\) is at \(x=-\frac{b}{2a}\). Substituting \(a = 2\) and \(b = 0\), we get \(x = 0\). But for the interval \([3,6]\):

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

When \(x = 3\), \(y=f(3)=2\times3^{2}+3=2\times9 + 3=18 + 3=21\).
When \(x = 6\), \(y=f(6)=2\times6^{2}+3=2\times36+3=72 + 3=75\).

Answer:

The absolute maximum value is \(75\) at \(x = 6\)