Question

Determine whether f(x) =1/2x2-x-9 has a maximum or a minimum value and find that value.

Answer 1

Given:

`f(x)=(1)/(2)x^2-x-9`

Differentiate with respect to 'x'

`\begin{gathered} f^(\prime)(x)=(1)/(2)(2x)-1 \\ f^(\prime)(x)=x-1 \end{gathered}`

Let f'(x)=0

`\begin{gathered} x-1=0 \\ x=1 \end{gathered}`
`\begin{gathered} f^(\doubleprime)(x)=1 \\ f^(\doubleprime)(x)>0 \end{gathered}`

Therefore, the function f(x) is minimum

`\begin{gathered} f(1)=(1)/(2)(1)^2-1-9 \\ f(1)=(1)/(2)-1-9 \\ f(1)=(1-2-18)/(2) \\ f(1)=-(19)/(2) \\ f(1)=-9.5 \end{gathered}`
`\text{Therefore, the minimum point is (1,-9.5)}`

Option d is the final answer.