use the function f(x)=2x^4 - 16x^2 - 7 to complete all parts of this question. round your answers to two…

use the function f(x)=2x^4 - 16x^2 - 7 to complete all parts of this question. round your answers to two decimal places. (a) this function has three critical numbers (cns) which are: cn1 is x = (smallest) cn2 is x = (middle) cn3 is x = (largest) (b) find the absolute maximum and minimum values of f(x)on the interval -1≤x≤4. the absolute maximum value of f(x) is and it occurs at x = the absolute minimum value of f(x) is and it occurs at x =

use the function f(x)=2x^4 - 16x^2 - 7 to complete all parts of this question. round your answers to two decimal places. (a) this function has three critical numbers (cns) which are: cn1 is x = (smallest) cn2 is x = (middle) cn3 is x = (largest) (b) find the absolute maximum and minimum values of f(x)on the interval -1≤x≤4. the absolute maximum value of f(x) is and it occurs at x = the absolute minimum value of f(x) is and it occurs at x =

Answer

Explanation:

Step1: Find the derivative of $f(x)$

Using the power - rule $(x^n)'=nx^{n - 1}$, if $f(x)=2x^{4}-16x^{2}-7$, then $f'(x)=8x^{3}-32x = 8x(x^{2}-4)=8x(x - 2)(x + 2)$.

Step2: Find the critical numbers

Set $f'(x)=0$. Then $8x(x - 2)(x + 2)=0$. Solving this equation gives $x=-2,0,2$. So $CN_1=-2$, $CN_2 = 0$, $CN_3=2$.

Step3: Evaluate $f(x)$ at critical numbers and endpoints

  • Evaluate $f(x)$ at $x=-2$: $f(-2)=2(-2)^{4}-16(-2)^{2}-7=2\times16-16\times4 - 7=32-64 - 7=-39$.
  • Evaluate $f(x)$ at $x = 0$: $f(0)=2(0)^{4}-16(0)^{2}-7=-7$.
  • Evaluate $f(x)$ at $x = 2$: $f(2)=2(2)^{4}-16(2)^{2}-7=32-64 - 7=-39$.
  • Evaluate $f(x)$ at $x=-1$: $f(-1)=2(-1)^{4}-16(-1)^{2}-7=2 - 16 - 7=-21$.
  • Evaluate $f(x)$ at $x = 4$: $f(4)=2(4)^{4}-16(4)^{2}-7=2\times256-16\times16 - 7=512-256 - 7=249$.

Step4: Determine the absolute maximum and minimum

The largest value among the above - calculated values is $249$ which occurs at $x = 4$, and the smallest value is $-39$ which occurs at $x=\pm2$.

Answer:

(a) $CN_1$ is $x=-2$ (smallest) $CN_2$ is $x = 0$ (middle) $CN_3$ is $x=2$ (largest) (b) The absolute maximum value of $f(x)$ is $249$ and it occurs at $x = 4$. The absolute minimum value of $f(x)$ is $-39$ and it occurs at $x=-2$ and $x = 2$.