Find the value or values of $c$ that satisfy the equation $\frac { f ( b ) - f ( a ) } { b - a } = f ^ { \prime } ( c )$ in the conclusion of the Mean Value Theorem for the function and interval.
-$$f ( x ) = x ^ { 2 } + 5 x + 2 , [ 1,2 ]$$
A) $- \frac { 3 } { 2 } , \frac { 3 } { 2 }$
B) 1,2
C) $\frac { 3 } { 2 }$
D) $0 , \frac { 3 } { 2 }$

Question

Quizplus · Accepted Answer

Applying the Mean Value Theorem for the given function $f(x)=x^2+5x+2$ and interval $[1,2]$ we have, 
$$\frac{f(2)-f(1)}{2-1}=f'(c)$$
$$\Rightarrow f(2)-f(1)=f'(c)$$ 
Now, we have to find the value of $c$. So, let's calculate $f'(x)$ 
$$f(x)=x^2+5x+2$$
$$\Rightarrow f'(x)=2x+5$$
Let's substitute $c$ in $f'(x)$
$$f'(c)=2c+5$$
So, we need to find the value of $c$ which makes 
$$f(2)-f(1)= f'(c)$$ 
$$\Rightarrow (2^2+5*2+2)-(1^2+5*1+2)=2c+5$$ 
$$\Rightarrow 12=2c+5$$ 
$$\Rightarrow c=\frac{7}{2}$$

Hence, option C is the correct choice.

Find the Value or Values Of ccc That Satisfy the Equation f(b)−f(a)b−a=f′(c)\frac { f ( b ) - f ( a ) } { b - a } = f ^ { \prime } ( c )b−af(b)−f(a)​=f′(c)

Find the Value or Values Of $c$ That Satisfy the Equation $\frac { f ( b ) - f ( a ) } { b - a } = f ^ { \prime } ( c )$