find the absolute maximum and minimum values of the function over the indicated interval.\nf(x)=2x² +…

find the absolute maximum and minimum values of the function over the indicated interval.\nf(x)=2x² + 3\n(a)3,6 (b)-6,6\n(a) the absolute maximum value is 75 at x = 6.\n(use a comma to separate answers as needed.)\nthe absolute minimum value is 21 at x = 3.\n(use a comma to separate answers as needed.)\n(b) the absolute maximum value is □ at x = □.\n(use a comma to separate answers as needed)

find the absolute maximum and minimum values of the function over the indicated interval.\nf(x)=2x² + 3\n(a)3,6 (b)-6,6\n(a) the absolute maximum value is 75 at x = 6.\n(use a comma to separate answers as needed.)\nthe absolute minimum value is 21 at x = 3.\n(use a comma to separate answers as needed.)\n(b) the absolute maximum value is □ at x = □.\n(use a comma to separate answers as needed)

Answer

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).