Question

1. For the function f, given by \[f\left( x \right)=2{{x}^{3}}-9{{x}^{2}}+12x+9\] , show that \[f'\left( 1 \right)=f'\left( 2 \right)\]
2. For the function f, given by \[f\left( x \right)={{x}^{2}}-6x-7\] , show that \[f'\left( 5 \right)-3f'\left( 2 \right)=f'\left( 8 \right)\].

Answer 1

Hint: In this problem, we have to prove the given conditions to be equal or not. We are given an equation in two problems, where we have to differentiate them and substitute the given value in the question and we have to equate them to find whether the given condition is correct. Here we can differentiate the given equation using differentiation formulas.

Complete step by step answer:
1.Here we have to prove \[f'\left( 1 \right)=f'\left( 2 \right)\], if \[f\left( x \right)=2{{x}^{3}}-9{{x}^{2}}+12x+9\].
We can now find \[f'\left( x \right)\] by differentiating the given equation, we get
 \[\begin{align}
  & \Rightarrow f'\left( x \right)=3\times 2{{x}^{2}}-2\times 9{{x}^{1}}+12 \\
 & \Rightarrow f'\left( x \right)=6{{x}^{2}}-18x+12..........(1) \\
\end{align}\]
We can now find the \[f'\left( 1 \right)\] by substituting x = 1 in the above step, we get
\[\Rightarrow f'\left( 1 \right)=6\left( 1 \right)-18\left( 1 \right)+12=0\]……. (2)
We can now find \[f'\left( 2 \right)\] by substituting x = 2 in (1), we get
\[\Rightarrow f'\left( 2 \right)=6\left( 4 \right)-18\left( 2 \right)+12=0.......(3)\]
We can see that from (2) and (3), \[f'\left( 1 \right)=f'\left( 2 \right)\].
Hence proved.
2. Here Here we have to prove \[f'\left( 5 \right)-3f'\left( 2 \right)=f'\left( 8 \right)\], if \[f\left( x \right)={{x}^{2}}-6x-7\].
We can now find \[f'\left( x \right)\] by differentiating the given equation, we get
 \[\Rightarrow f'\left( x \right)=2{{x}^{1}}-6.......(4)\]
We can now find the \[f'\left( 5 \right)\] by substituting x = 5 in the above step, we get
\[\Rightarrow f'\left( 5 \right)=2\left( 5 \right)-6=4\]……. (5)
We can now find \[3f'\left( 2 \right)\] by substituting x = 2 in (4), we get
\[\Rightarrow 3f'\left( 2 \right)=3\left( 2\left( 2 \right)-6 \right)=-6.......(6)\]
We can now find \[f'\left( 8 \right)\] by substituting x = 8 in (4), we get
\[\Rightarrow f'\left( 8 \right)=2\left( 8 \right)-6=10.......(7)\]
We can now substitute (5), (6), (7) in \[f'\left( 5 \right)-3f'\left( 2 \right)=f'\left( 8 \right)\], we get
\[\Rightarrow 4-\left( -6 \right)=10\]
Therefore, \[f'\left( 5 \right)-3f'\left( 2 \right)=f'\left( 8 \right)\].
Hence proved.

Note: We should always remember some of the differentiating formulas such as \[{{x}^{2}}dx=2x\]. We should also remember that if we differentiate a constant term then we will get 1 as the answer. We should also know that \[f'\left( x \right)\] is nothing but a differentiation of \[f\left( x \right)\].