Questions: Find the average value (ftext ave ) of (f(x)=x^7) between -1 and 1 , then find a number (c) in ([-1,1]) where (f(c)=ftext ave ).

Solution Steps

To find the average value \( f_{\text{ave}} \) of the function \( f(x) = x^7 \) over the interval \([-1, 1]\), we use the formula:

\[ f_{\text{ave}} = \frac{1}{b-a} \int_{a}^{b} f(x) \, dx \]

Substituting \( a = -1 \) and \( b = 1 \):

\[ f_{\text{ave}} = \frac{1}{1 - (-1)} \int_{-1}^{1} x^7 \, dx \]

Calculating the integral:

\[ \int_{-1}^{1} x^7 \, dx = 0 \]

Thus, we find:

\[ f_{\text{ave}} = \frac{1}{2} \cdot 0 = 0 \]

Next, we need to find a number \( c \) in the interval \([-1, 1]\) such that \( f(c) = f_{\text{ave}} \). Since we have determined that \( f_{\text{ave}} = 0 \), we set up the equation:

\[ f(c) = c^7 = 0 \]

To solve for \( c \), we find the values that satisfy the equation:

\[ c^7 = 0 \]

The solution to this equation is:

\[ c = 0 \]

Thus, the value of \( c \) in the interval \([-1, 1]\) where \( f(c) = f_{\text{ave}} \) is \( c = 0 \).

Final Answer