Question

How to find (f) if f′(x)=16x^3+6x+7 and f(1)=−1 ?

Answer 1

`f(x) = 4x^4 + 3x^2 + 7x - 15`

Explanation:

We start by integrating both sides.

`f(x) = int 16x^3 + 6x + 7 dx`

`f(x) = 4x^4 + 3x^2 + 7x + C`

Now we solve for `C`.

`-1 = 4(1)^4 + 3(1)^2 + 7(1) + C`

`-1 = 4 + 3 + 7 + C`

`-1 - 14 = C`

`C = -15`

Hopefully this helps!

Answer 2

`f(x)=4x^4+3x^2+7x-15`

Explanation:

Integrate:
`16x^3` becomes `4x^4`
`6x` becomes `3x^2`
`7` becomes `7x`

So `f(x)` is `4x^4+3x^2+7x+C`.
Plug in `x=1`:

`4(1^4)+3(1^2)+7(1)+C`
`=4+3+7+C`
`=14+C`

Set `14+C` equal to `-1`:
`-1=14+C`

Solve for C:
`-15=C`

So your `f(x) = 4x^4 + 3x^2+7x-15`

Answer 3

`f(x)=4x^4+3x^2+7x-15`

Explanation:

We got:

`f'(x)=16x^3+6x+7`

`f(1)=-1`

And so,

`f(x)=intf'(x) \ dx`

`=int16x^3+6x+7 \ dx`

`=4x^4+3x^2+7x+C`

Therefore,

`4(1)^4+3(1)^2+7*1+C=-1`

`4*1+3*1+7+C=-1`

`4+3+7+C=-1`

`14+C=-1`

`C=-15`

So, the original function `f(x)` is:

`color(blue)(f(x)=barul|4x^4+3x^2+7x-15|)`