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 75 at x = 6.
(use a comma to separate answers as needed.)
the absolute minimum value is 21 at x = 3.
(use a comma to separate answers as needed.)
(b) the absolute maximum value is □ at x = □.
(use a comma to separate answers as needed)

Explanation:

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

The function \(y = f(x)=2x^{2}+3\) 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 \(x=-\frac{b}{2a}\). For \(y = 2x^{2}+3\), \(x = 0\) (because \(b = 0\) and \(a = 2\)). The value of the function at \(x = 0\) is \(f(0)=2\times0^{2}+3=3\).

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

• When \(x=-6\), \(f(-6)=2\times(-6)^{2}+3=2\times36 + 3=72 + 3=75\).
• When \(x = 6\), \(f(6)=2\times6^{2}+3=2\times36+3=72 + 3=75\).

Step3: Compare the values

We have \(f(0) = 3\), \(f(-6)=75\), \(f(6)=75\).

Answer:

The absolute maximum value is \(75\) at \(x=-6,6\). The absolute minimum value is \(3\) at \(x = 0\).