Answer:
[tex] \displaystyle - 155[/tex]
Step-by-step explanation:
we are given a quadratic function
[tex] \displaystyle f(x) = - 5 {x}^{2} + 30x - 200[/tex]
we want to figure out the minimum value of the function
to do so we need to figure out the minimum value of x in the case we can consider the following formula:
[tex] \displaystyle x _{ \rm min} = \frac{ - b}{2a} [/tex]
the given function is in the standard form i.e
[tex] \displaystyle f(x) = a {x}^{2} + bx + c[/tex]
so we acquire:
thus substitute:
[tex] \displaystyle x _{ \rm min} = \frac{ - 30}{2. - 5} [/tex]
simplify multiplication:
[tex] \displaystyle x _{ \rm min} = \frac{ - 30}{ - 10} [/tex]
simply division:
[tex] \displaystyle x _{ \rm min} = 3[/tex]
plug in the value of minimum x to the given function:
[tex] \displaystyle f (3)= - 5 {(3)}^{2} + 30.3 - 200[/tex]
simplify square:
[tex] \displaystyle f (3)= - 5 {(9)}^{} + 30.3 - 200[/tex]
simplify multiplication:
[tex] \displaystyle f (3)= - 45 + 90- 200[/tex]
simplify:
[tex] \displaystyle f (3)= - 155[/tex]
hence,
the minimum value of the function is -155
