Question

The value of $x$ for which the value of the function $f\left( x \right)=-\dfrac{11}{3}{{x}^{2}}+17x-\dfrac{43}{13}$ is maximum is
(A) $x=-\dfrac{102}{11}$
(B) $x=-\dfrac{51}{22}$
(C) $x=\dfrac{51}{22}$
(D) $x=\dfrac{102}{11}$

Answer 1

Hint: For answering this question we need to find the value of $x$ for which the value of the function $f\left( x \right)=-\dfrac{11}{3}{{x}^{2}}+17x-\dfrac{43}{13}$ is maximum. From the basic concept we know that the maximum of a function occurs when its derivative is zero.

Complete step by step answer:
Now from the question we have the function $f\left( x \right)=-\dfrac{11}{3}{{x}^{2}}+17x-\dfrac{43}{13}$ .
The value of $x$ for which the value of $f\left( x \right)$ is maximum, $f'\left( x \right)$ should be zero.
So we need to differentiate the function.
For differentiating the function we will use the formulae
$\dfrac{d}{dx}a{{x}^{-2}}=-2ax$ . After applying this formula in the equation we have we will get
${{f}^{'}}\left( x \right)=-\dfrac{11}{3}\left( 2 \right)x+17$ .
By equating it to zero we will have,
${{f}^{'}}\left( x \right)=-\dfrac{11}{3}\left( 2 \right)x+17=0$
By simplifying this equation we will have,
$-11\left( 2 \right)x+17\left( 3 \right)=0$
After further simplifications we will have,
$\begin{align}
  & -22x+51=0 \\
 & \Rightarrow 22x=51 \\
 & \Rightarrow x=\dfrac{51}{22} \\
\end{align}$.
For a value of $x$ the function $f\left( x \right)$ will have maximum or minimum value only when its derivative is equal to zero that is mathematically given as $f'\left( x \right)=0$ and for the minimum of the function its double derivative will be positive given as $f''(x)>0$ and for maximum value it will be negative given as $f''\left( x \right)<0$ .
Here now we have the value of $x$ for which the first derivative of $f\left( x \right)$ is zero. We need the value of $x$ for which it is maximum and it occurs when the second derivative is less than zero. Let us verify this.
The second derivative of $f\left( x \right)$ is given as $f''\left( x \right)$. We have the first derivative as $f'\left( x \right)=-\dfrac{11}{3}2x+17$ . The derivative of this is given as $f''\left( x \right)=-\dfrac{22}{3}$ . For the value of $x$, for which is negative it indicates that the value of $f\left( x \right)$ will be maximum.
We had come to a conclusion after performing the simplifications and shifting the values from the Left hand side to right hand side.
The conclusion we have got from this is the value of $x$ for which the value of the function $f\left( x \right)=-\dfrac{11}{3}{{x}^{2}}+17x-\dfrac{43}{13}$ is maximum is given as $x=\dfrac{51}{22}$ .

So, the correct answer is “Option C”.

Note: While answering this type of questions we need to be clear with calculations and we should remember a further point which states that for a value of $x$ the function $f\left( x \right)$ will have maximum or minimum value only when its derivative is equal to zero that is mathematically given as $f'\left( x \right)=0$ and for the maximum of the function its double derivative will be negative given as $f''(x)<0$ and for minimum value it will be positive given as $f''\left( x \right)>0$ .