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 following function and interval.
\[
f(x)=3 x^{2}+5 x-2 \quad[-1,0]
\]
Answer
Finally, we set the derivative equal to the average rate of change and solve for \(c\). This gives us \(c = -\frac{1}{2}\).
Step 1 ：Given the function \(f(x)=3x^2+5x-2\) and the interval [-1,0], we need to find the value of \(c\) that satisfies the equation \(\frac{f(b)-f(a)}{b-a}=f^{\prime}(c)\) according to the Mean Value Theorem.
Step 2 ：First, we find the derivative of the function, \(f^{\prime}(x)\), which is \(6x + 5\).
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 \(c = -\frac{1}{2}\).
