Question

determine whether each function has a maximum or a minimum value. then find the maximum or minimum value of each function.

1. ( f(x) = x^2 + 2x - 8 )
2. ( v(x) = -x^2 + 14x - 57 )

Explanation:

Step1: Analyze $f(x)=x^2+2x-8$

For a quadratic $ax^2+bx+c$, $a=1>0$, so it has a minimum.

Step2: Find vertex x-coordinate

$x=-\frac{b}{2a}=-\frac{2}{2(1)}=-1$

Step3: Calculate minimum of $f(x)$

$f(-1)=(-1)^2+2(-1)-8=1-2-8=-9$

Step4: Analyze $v(x)=-x^2+14x-57$

$a=-1<0$, so it has a maximum.

Step5: Find vertex x-coordinate

$x=-\frac{b}{2a}=-\frac{14}{2(-1)}=7$

Step6: Calculate maximum of $v(x)$

$v(7)=-(7)^2+14(7)-57=-49+98-57=-8$

Answer:

1. For $f(x)=x^2+2x-8$: It has a minimum value, and the minimum value is $-9$.
2. For $v(x)=-x^2+14x-57$: It has a maximum value, and the maximum value is $-8$.