Question

Let $f\left(x\right)$ be a differential function and satisfy $f\left(0\right)=2$, and $f^{\prime }\left(x\right)=f\left(x\right)$ . Find

• A. The range of the function $f\left(x\right)is(0,\infty )$
• B. The value of the function when $x=ln2is3$
• C. The area enclosed by $y=f\left(x\right)$ in the $2nd$ quadrant is bounded.
• D. None of these

Answer 1

The correct option is A The range of the function $f\left(x\right)is(0,\infty )$

$f^{{}^{\prime }}\left(x\right)=f\left(x\right)$

$\Rightarrow \frac{dy}{dx}=y$

$\Rightarrow \frac{dy}{y}=dx$

Now on integrating, we get

$\log y=x+C$

$f\left(0\right)=2\Rightarrow C=\log 2$

$\log y=x+C=x+\log 2$

$y=2e^x$

Therefore, we can say that range of f(x) is $(0,\infty )$.