# How do you find the point c in the interval #-4<=x<=6# such that f(c) is equation to the average value of #f(x)=2x#?

For our case we get

Solve:

Evaluating the integral we get:

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To find the point ( c ) in the interval ([-4, 6]) such that ( f(c) ) is equal to the average value of ( f(x) = 2x ) over that interval, you first need to find the average value of ( f(x) ) over the interval.

The average value of a function ( f(x) ) over an interval ([a, b]) is given by:

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

For ( f(x) = 2x ), the interval is ([-4, 6]), so:

[ \text{Average value} = \frac{1}{6 - (-4)} \int_{-4}^{6} 2x , dx ]

[ = \frac{1}{10} \int_{-4}^{6} 2x , dx ]

[ = \frac{1}{10} [x^2]_{-4}^{6} ]

[ = \frac{1}{10} [(6)^2 - (-4)^2] ]

[ = \frac{1}{10} [36 - 16] ]

[ = \frac{1}{10} \cdot 20 ]

[ = 2 ]

So, the average value of ( f(x) ) over ([-4, 6]) is 2. To find the point ( c ) where ( f(c) ) is equal to this average value, set ( f(c) = 2x ) and solve for ( c ):

[ 2c = 2 ]

[ c = 1 ]

Therefore, the point ( c ) in the interval ([-4, 6]) such that ( f(c) ) is equal to the average value of ( f(x) = 2x ) is ( c = 1 ).

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