find the absolute maximum and minimum values of the function over the indicated interval. f(x)=2x^2 + 5 (a)…

find the absolute maximum and minimum values of the function over the indicated interval. f(x)=2x^2 + 5 (a) 3,6 (b) -6,6 (a) the absolute maximum value is 77 at x = 6. (use a comma to separate answers as needed.) the absolute minimum value is at x =. (use a comma to separate answers as needed.)

find the absolute maximum and minimum values of the function over the indicated interval. f(x)=2x^2 + 5 (a) 3,6 (b) -6,6 (a) the absolute maximum value is 77 at x = 6. (use a comma to separate answers as needed.) the absolute minimum value is at x =. (use a comma to separate answers as needed.)

Answer

Explanation:

Step1: Analyze the function

The function $f(x)=2x^{2}+5$ is a quadratic function. Its graph is a parabola opening upwards ($a = 2>0$ in the general form $y = ax^{2}+bx + c$, here $b = 0,c = 5$).

Step2: Evaluate the function at endpoints for interval [3,6]

For $x = 3$, $f(3)=2\times3^{2}+5=2\times9 + 5=18 + 5=23$. For $x = 6$, $f(6)=2\times6^{2}+5=2\times36+5 = 72 + 5=77$. Since the function is increasing on the interval $[3,6]$ (because the parabola $y = 2x^{2}+5$ is increasing for $x\geq0$), the minimum value on $[3,6]$ occurs at $x = 3$.

Answer:

23,3