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

Hint:
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.