Question

Find a function $f$that has the given derivative $f\text{'}$and value f(c).

$f\text{'}\left(x\right)=-32x+4,f\left(0\right)=100$

Short answer

Antiderivaive$f\left(x\right)=-16x^2+4x$$+100$

Step-by-step solution

Step 1:Given information

Given expression is$f\text{'}\left(x\right)=-32x+4,f\left(0\right)=100$

Step 2:Simplification

given

$f\text{'}\left(x\right)=-32x+4$

We will use an intelligent guess-and-check method to find f

here,$-32x$is derivative of $-16x^2$and 4 is derivaive of $4x$

hence,

$f\left(x\right)=-16x^2+4x$$+c$

now

$$\begin{gathered} f\left(0\right)=100, \\ \Rightarrow 100=0+c \end{gathered}$$

$\Rightarrow c=100$

Therefore

$f\left(x\right)=-16x^2+4x$$+100$