Find the value or values of c that satisfy (f(b)-f(a))/(b-a) =f'(c)for the function f(x)=x^2+3x+3 on the interval [-2,1]. I can't seem to get -1/2, the answer given.

[((1)^2+3(1)+3)-((-2)^2+3(-2)+3)]/[1+2]
=2

I just set the 2= the derivative of the function (which is 2c+3) and solve for c. I just figured it out. No help needed here.

To find the value or values of c that satisfy the equation

(f(b) - f(a))/(b - a) = f'(c)

for the function f(x) = x^2 + 3x + 3 on the interval [-2, 1], we can start by calculating f(b), f(a), and f'(c) separately.

Let's substitute b = 1 and a = -2 into the function f(x):

f(b) = f(1) = (1)^2 + 3(1) + 3 = 1 + 3 + 3 = 7
f(a) = f(-2) = (-2)^2 + 3(-2) + 3 = 4 - 6 + 3 = 1

Now, let's calculate the derivative, f'(c), by differentiating f(x) with respect to x:

f'(x) = 2x + 3

Since we are looking for the value or values of c on the interval [-2, 1], we need to find a value of c that lies within this range.

Now, let's set up the equation:

(f(b) - f(a))/(b - a) = f'(c)

(7 - 1)/(1 - (-2)) = 2c + 3

Now simplify the left side of the equation:

6/3 = 2c + 3

2 = 2c + 3

Subtract 3 from both sides:

-1 = 2c

Now, divide both sides by 2:

-1/2 = c

Therefore, c = -1/2 is the value of c that satisfies the equation on the interval [-2, 1].

It seems there was an error in your calculation. Let's review your calculation step by step:

[((1)^2+3(1)+3)-((-2)^2+3(-2)+3)]/[1+2]
=(1+3+3-(-2+6+3))/3
=(7-(-2+6+3))/3
=(7-7)/3
=0/3
=0

It seems you made an error in calculating f(b) and f(a). The correct values for f(b) and f(a) are 7 and 1, respectively, as calculated above.

Thus, the correct value of c that satisfies the equation is -1/2.