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

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Finally, we set the derivative equal to the average rate of change and solve for $c$ . This gives us $c = - \frac{1}{2}$ .

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Step 1 ：Given the function $f (x) = 3 x^{2} + 5 x - 2$ and the interval [-1,0], we need to find the value of $c$ that satisfies the equation $\frac{f (b) - f (a)}{b - a} = f^{'} (c)$ according to the Mean Value Theorem.

Step 2 ：First, we find the derivative of the function, $f^{'} (x)$ , which is $6 x + 5$ .

Step 3 ：Next, we calculate the average rate of change of the function over the interval, which is $\frac{f (b) - f (a)}{b - a}$ . Substituting $a = - 1$ and $b = 0$ into the function, we get an average rate of change of 2.

Step 4 ：Finally, we set the derivative equal to the average rate of change and solve for $c$ . This gives us $c = - \frac{1}{2}$ .

Find the value or values of c that satisfy the equation f(b)−f(a)b−a=f′(c) in the conclusion of the Mean Value Theorem for the following function and interval.f(x)=3x2+5x−2[−1,0]

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