To find the area under the curve, we need to integrate. In this case, it is a definite integral.

\(\displaystyle \int_{2}^{7}\frac{1}{\sqrt{x+2}}dx=2\sqrt{x+2}\Big|_2^7=2\)

The easiest way to look at this is to plot the graphs. The shaded area is the actual area that we want to compute. We can first find area bounded by \(\displaystyle x=3\) and \(\displaystyle y=2\) in the first quadrant and subtract the excessive areas. The area of that rectangle box is 6. The area under the curve \(\displaystyle \frac{1}{9}x^{2}\) is \(\displaystyle \int_{0}^{3}\frac{1}{9}x^2=1\).

The area of the triangle above the curve \(\displaystyle y=x\) is 2. Therefore, the area bounded is \(\displaystyle 6-1-2=3\).

\(\displaystyle \int sin(x) dx = -cos(x) + C\)\(\displaystyle \int _0^\pi sin(x) dx = -cos(\pi) - -cos(0)= 1-(-1) = 2\)

The area of this region is given by the following integral:

\(\displaystyle Area = \int_{0}^{2} f(x) - g(x) = \int_{0}^{2} -x^{2} + 2x - (-sin(\frac{\pi x}{2}))\) or

\(\displaystyle Area = \int_{0}^{2} -x^{2} + 2x + sin(\frac{\pi x}{2})\)

Taking the antiderivative gives

\(\displaystyle -\frac{x^{3}}{3} + x^{2} -\frac{2}{\pi }cos(\frac{\pi x}{2})\), evaluated from \(\displaystyle x = 0\) to \(\displaystyle x = 2\).

\(\displaystyle F(2) = \frac{-8}{3} + 4 + \frac{2 }{\pi }\), and 

\(\displaystyle F(0) = -\frac{2}{\pi }\).

Thus, the area is given by:

\(\displaystyle F(2) - F(0) =- \frac{8}{3} + 4 + \frac{2}{\pi } - (-\frac{2}{\pi }) =\frac{4}{3} +\frac{4}{\pi }\)

By Rolle's Theorem, if \(\displaystyle f\) is continuous on \(\displaystyle [a,b]\) and differentiable on \(\displaystyle (a,b)\), and \(\displaystyle f(a)= f(b)\), then there must be \(\displaystyle c \in \left ( a,b \right )\) such that \(\displaystyle f'(c) = 0\). Nothing in the statement of this theorem addresses the location of the zeroes of the function itself. Therefore, the statement is false.

By the Mean Value Theorem (MVT), if a function \(\displaystyle f\) is continuous and differentiable on \(\displaystyle [a,b]\), then there exists at least one value \(\displaystyle c \in (a,b)\) such that \(\displaystyle f'(c) = \frac{f(b)-f(a)}{b-a}\). \(\displaystyle f\), a polynomial, is continuous and differentiable everywhere; setting \(\displaystyle a= -2, b= 2\), it follows from the MVT that there is \(\displaystyle c \in (-2, 2 )\) such that 

\(\displaystyle f'(c) = \frac{f(2)-f(-2)}{2-(-2)}\)

Evaluating \(\displaystyle f(-2)\) and \(\displaystyle f(2)\):

\(\displaystyle f(x) = x^{3} - 7x + 16\)

\(\displaystyle f(2) = 2^{3} - 7 (2) + 16 = 8 -14+16 = 10\)

\(\displaystyle f(-2) = (-2) ^{3} - 7 (-2) + 16 = -8 +14+16 = 22\)

The expression for \(\displaystyle f'(c)\) is equal to 

\(\displaystyle f'(c) = \frac{10-22}{2-(-2)} = \frac{-12}{4} = -3\),

the correct choice.

By Rolle's Theorem, if \(\displaystyle f\) is continuous on \(\displaystyle [a,b]\) and differentiable on \(\displaystyle (a,b)\), and \(\displaystyle f(a)= f(b)\), then there must be \(\displaystyle c \in \left ( a,b \right )\) such that \(\displaystyle f'(c) = 0\). 

\(\displaystyle f\) is given to be continuous. Also, if we set \(\displaystyle a=1, b=2\), we note that \(\displaystyle f(1) = f(2)=0\). This sets up the conditions for Rolle's Theorem to apply. As a consequence, there must be \(\displaystyle c \in (1,2)\) such that \(\displaystyle f'(c) = 0\).

Incidentally, it does follow from the given information that \(\displaystyle f(x)\) must have a zero on the interval \(\displaystyle (3,4)\), but this is due to the Intermediate Value Theorem, not Rolle's Theorem.

To find the mean value of a function over some interval \(\displaystyle (a,b)\), one mus use the formula: \(\displaystyle (f(b)-f(a))=(b-a)f'(c)\).

Plugging in 

\(\displaystyle (f(\frac{\pi}{2})-f(0))=(\frac{\pi}{2})(-Sin(c))\)

Simplifying

\(\displaystyle -1=\frac{\pi}{2}(-Sin(c))\)

\(\displaystyle \frac{2}{\pi}=Sin(c)\)

One must then use the inverse Sine function to find the value c:

\(\displaystyle c=sin^{-1}\frac{2}{\pi}\)