Question

The graph of y = f ′(x), the derivative of f(x), is shown below. Given f(4) = 6, evaluate f(0).

A. -2
B. 2
C. 4
D. 10

Answer 1

B. 2

Step-by-step explanation:

From the graph we get

(1).
`f'(x)=x` for
`-2\leq x\leq 2`

(2).
`f'(x)=-x+4` for
`x>2`

(3).
`f'(x)=-x-4` for
`x<-2`

Now let us take the anti derivative of the second function, and we get

`f(x)=-(x^2)/(2) +4x+c` for
`x>2`

We find
`c` from the condition
`f(4)=6`:

`f(4)=-(4^2)/(2) +(4*4)+c=6`

`8+c=6`

`\boxed{c=-2}`

Thus we have

`f_2(x)=-(x^2)/(2) +4x-2`.

Now we find the anti derivative of the first function, and get

`f_1(x)=(x^2)/(2) +c_1`

What we note now is that the function is continuous because the derivative of the functions is defined at
`x=2` and at
`x=-2`.

since the function are continuous,
`f_1(2)=f_2(2)`

`-(2^2)/(2) +4(2)-2=((2)^2)/(2) +c_1`

`4=2+c`

`\boxed{c_1=2}`

Thus we have

`f_1(x)=(x^2)/(2) +2`

Now we can easily evaluate
`f(0)`

`f_1(0)=0+2=2\\\\\boxed{f(0)=2}`

Which is choice B.