Question

If $f\left( x \right) = {e^x}g\left( x \right)$, $g\left( 0 \right) = 2$, $g'\left( 0 \right) = 1$, then $f\left( 0 \right)$ is
A) 1
B) 2
C) 3
D) 4

Answer 1

Here, we will first replace 0 for $x$ in the given equation and then we will substitute the value of $g\left( 0 \right)$ in the obtained equation to find the value of $f\left( 0 \right)$ for the required value.

Complete step by step solution:
We are given that the equation is $f\left( x \right) = {e^x}g\left( x \right)$.
Replacing 0 for $x$ in the above equation, we get

$$\Rightarrow f\left( 0 \right) = {e^0}g\left( 0 \right) \\\ \Rightarrow f\left( 0 \right) = 1 \times g\left( 0 \right) \\\ \Rightarrow f\left( 0 \right) = g\left( 0 \right) \\\$$

Substituting the value of $g\left( 0 \right)$ in the above equation, we get
$\Rightarrow f\left( 0 \right) = 2$

Hence, option B is correct.

Note:
In solving these types of questions, students should know to use the values of given conditions. We should know that the value of $g'\left( 0 \right) = 1$ is not used in the solution of this problem. Using this value will only lead to confusion to a student. We have to be careful while plugging in the known value of ${e^x}$ for $x = 0$ negates the exponential function.