find the absolute maximum and minimum values of the function over the indicated interval.\n\n$f(x)=2…

find the absolute maximum and minimum values of the function over the indicated interval.\n\n$f(x)=2 x^{2}+3$\n(a) $3,6$\n(b) $-6,6$\n\n(a) the absolute maximum value is 75 at $x=6$\n(use a comma to separate answers as needed)\n\nthe absolute minimum value is $\\square$ at $x=\\square$\n(use a comma to separate answers as needed)

find the absolute maximum and minimum values of the function over the indicated interval.\n\n$f(x)=2 x^{2}+3$\n(a) $3,6$\n(b) $-6,6$\n\n(a) the absolute maximum value is 75 at $x=6$\n(use a comma to separate answers as needed)\n\nthe absolute minimum value is $\\square$ at $x=\\square$\n(use a comma to separate answers as needed)

Answer

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)