# Given f ''(x) = 6x - 4 and f '(0)=1 and f(0)=5 find f '(x) and f (1)

## Question:

Given {eq}f ''(x) = 6x - 4, \; f '(0)=1, \; \text{and} \; f(0)=5, \; \text{find} \; f '(x) \; \text{and} \; f (1) {/eq}

## Using Integration to Find a Function

We are given the second derivative of a function as well as the values of the first derivative of the function and the function at fixed points. Using integration twice, we find the exact form of the function as well as the first derivative of the function and the function value at a different point on its domain.

Given {eq}f''(x)=6x-4 {/eq} we integrate both sides with respect to x to obtain

Using f'(0) = 1 in (1) gives us

{eq}f'(0)=1=3(0^2)-4(0)+C \implies C = 1. {/eq} which yields

Next integrating (2) with respect to x as well leads to

Using f(0) = 5 in (3) then leads to

{eq}f(0)=5=0^3-2(0^2)+0+D \implies D = 5. {/eq}

So from (3)

From (2), {eq}f'(x)=3x^2-4x+1 {/eq} and from (4), {eq}f(1)=1^3-2(1^2)+1+5 = 5. {/eq}