section 2.8: optimization (homework)\nscore: 69.5/120 answered: 6/12\nquestion 8\nconsider the function…

Explanation:

Step1: Find the derivative of the function

Differentiate $f(x)=x^{4}-50x^{2}+1$ using the power - rule. $f^\prime(x)=4x^{3}-100x = 4x(x^{2}-25)=4x(x - 5)(x + 5)$.

Step2: Find the critical points

Set $f^\prime(x)=0$. Then $4x(x - 5)(x + 5)=0$. The critical points are $x=-5,x = 0,x = 5$. But $x=-5$ is outside the interval $[-4,11]$, so we discard it. We are left with $x = 0$ and $x = 5$.

Step3: Evaluate the function at the critical points and endpoints

Evaluate $f(x)$ at $x=-4,x = 0,x = 5,x = 11$. $f(-4)=(-4)^{4}-50(-4)^{2}+1=256-800 + 1=-543$. $f(0)=0^{4}-50\times0^{2}+1=1$. $f(5)=5^{4}-50\times5^{2}+1=625-1250 + 1=-624$. $f(11)=11^{4}-50\times11^{2}+1=14641-6050+1=8592$.

Answer:

The absolute minimum value is $-624$ and the absolute maximum value is $8592$.