Question

Given that \[\dfrac{d}{{dx}}f\left( x \right) = {f'}\left( x \right)\]. The relationship \[{f'}\left( {a + b} \right) = {f'}\left( a \right) + {f'}\left( b \right)\] is valid if \[f\left( x \right)\] is equal to
A. \[x\]
B. \[{x^2}\]
C. \[{x^3}\]
D. \[{x^4}\]

Answer 1

Hint: In this question, we are asked to check the relationship \[{f'}\left( {a + b} \right) = {f'}\left( a \right) + {f'}\left( b \right)\] is valid or not. For that, we will take the help of the given options and verify the condition.

Complete step-by-step solution:
We are given that \[\dfrac{d}{{dx}}f\left( x \right) = {f'}\left( x \right)\]
Now we check for the first option which is \[f(x) = x\]:
Now we take the derivative of the function, and we get
\[
  {f'}\left( x \right) = 1 \\
  {f'}\left( a \right) = 1 \\
  {f'}\left( b \right) = 1 \\
 \]
Now we substitute the values in the given relationship, and we get
\[
  {f'}\left( {a + b} \right) = {f'}\left( a \right) + {f'}\left( b \right) \\
   = 1 + 1 \\
   = 2 \\
 \]
Therefore, \[{f'}\left( x \right) = 1\] does not verify the given relationship condition.
Now we check for the second option which is \[f(x) = {x^2}\]:
Now we take the derivative of the function, and we get
\[
  {f'}\left( x \right) = 2x \\
  {f'}\left( a \right) = 2a \\
  {f'}\left( b \right) = 2b \\
 \]
Now we substitute the values in the given relationship, and we get
\[
  {f'}\left( {a + b} \right) = 2\left( {a + b} \right) \\
   = 2a + 2b \\
   = {f'}\left( a \right) + {f'}\left( b \right) \\
 \]
Therefore, option (B) satisfies the given relationship.
Now we check the third option which is \[f\left( x \right) = {x^3}\]:
Now we take the derivative of the function, and we get
\[
  {f'}\left( x \right) = 3{x^2} \\
  {f'}\left( a \right) = 3{a^2} \\
  {f'}\left( b \right) = 3{b^2} \\
 \]
Now we substitute the values in the given relationship, and we get
\[
  {f'}\left( {a + b} \right) = 3{\left( {a + b} \right)^2} \\
   = 3\left( {{a^2} + {b^2} + 2ab} \right)\left( {\because {{\left( {a + b} \right)}^2} = {a^2} + {b^2} + 2ab} \right) \\
   = 3{a^2} + 3{b^2} + 6ab \\
 \]
Therefore, option(C) does not satisfy the given relationship.
Now we check the fourth option which is \[f\left( x \right) = {x^4}\]:
Now we take the derivative of the function, and we get
\[
  {f'}\left( x \right) = 4{x^3} \\
  {f'}\left( a \right) = 4{a^3} \\
  {f'}\left( b \right) = 4{b^3} \\
 \]
Now we substitute the values in the given relationship, and we get
\[
  {f'}\left( {a + b} \right) = 4{\left( {a + b} \right)^3} \\
   = 4\left( {{a^3} + {b^3} + 3ab\left( {a + b} \right)} \right)\left( {\because {{\left( {a + b} \right)}^3} = {a^3} + {b^3} + 3ab\left( {a + b} \right)} \right) \\
   = 4{a^3} + 4{b^3} + 12ab\left( {a + b} \right) \\
 \]
Therefore, option(D) does not satisfy the given relationship.
Hence, option(B) is correct answer which satisfies the relationship \[{f'}\left( {a + b} \right) = {f'}\left( a \right) + {f'}\left( b \right)\]

Note: Students made mistake while taking differentiating of the function and substituting the values according to the relationship \[{f'}\left( {a + b} \right) = {f'}\left( a \right) + {f'}\left( b \right)\].