Question

If \[f\left( x \right) = m{x^2} + nx + p\], then \[{f^{'}}\left( 1 \right) + {f^{'}}\left( 4 \right) - {f^{'}}\left( 5 \right)\] is equal to
A. \[m\]
B. \[ - m\]
C. \[n\]
D. \[ - n\]

Answer 1

Hint: In this question, we are asked to find the value of \[{f^{'}}\left( 1 \right) + {f^{'}}\left( 4 \right) - {f^{'}}\left( 5 \right)\] . For that, we first take the derivative of the given function and then substitute the value \[ x= 1,4,5 \] to get the desired result.

Formula used:
1. \[\dfrac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}\]

Complete step-by-step solution:
We are given a function \[f\left( x \right) = m{x^2} + nx + p\] that is a second-degree polynomial Now we need to act on the first derivation, due to a small difference in the process of derivation, we call it a differentiation function. So, the differentiation formula is \[\dfrac{{dy}}{{dx}}\] .
\[{f^{'}}\left( x \right) = 2mx + n\]
Now substitute \[x = 1\], we obtain
\[{f^{'}}\left( 1 \right) = 2m + n\]
Now substitute \[x = 4\], we obtain
\[
  {f^{'}}\left( 4 \right) = 2m \times \,4 + n \\
   = 8m + n
 \]
Now substitute \[x = 5\], we obtain
\[
  {f^{'}}\left( 5 \right) = 2m \times \,5 + n \\
   = 10\,m + n
 \]
Thus, the required values of function \[{f^{'}}\left( 1 \right) + {f^{'}}\left( 4 \right) - {f^{'}}\left( 5 \right)\] are
\[
  {f^{'}}\left( 1 \right) + {f^{'}}\left( 4 \right) - {f^{'}}\left( 5 \right) = 2m + n + 8m + n - 10m - n \\
   = 2n - n \\
   = n
 \]
Hence, option (C) is correct

Note: Integration and differentiation are inverse processes. The derivative of x raised to the power is denoted by \[\dfrac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}\] in differentiation. Inverse differentiation is the process of integration. The process of determining which functions have a given derivative is known as anti-differentiation. It can calculate the area, volume, and central points. The anti-differentiation value at the upper limit and lower limit with the same anti-differentiation.