Question

If $f^{\prime }\left(x\right)=f\left(x\right)$ for all x and $f^{\prime }\left(0\right)=4$ then $f\left(x\right)$ is equal to

• A. $2e^{2x}$
• B. $e^{4x}$
• C. $x^4+4x^2+4x$
• D. $4e^x$

Answer 1

The correct option is D $4e^x$

given $f^1\left(x\right)=f\left(x\right)$

$\frac{df}{dx}=f$ which is equal to $\frac{df}{f}=dx$

now integrate on both sides , we get $\ln f\left(x\right)=x$

which implies $f\left(x\right)=k{e}^x$ where $k$ is constant.

given $f^1\left(0\right)=4=k$

so the equation is $f\left(x\right)=4e^x$