How do you find the point c in the interval #0<=x<=2# such that f(c) is equation to the average value of #f(x)=x^(2/3)#?

Answer 1

Solve the equation #f(x) = 1/(2-0) int_0^2 f(x) dx#. Discard any solutions outside #[0,2]#

#1/(2-0) int_0^2 f(x) dx = 1/2 int_0^2 x^(2/3) dx = (3(2^(2/3)))/5#

So you need to solve

#x^(2/3) = (3(2^(2/3)))/5#.
I get #x = (6sqrt15)/25#
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Answer 2

To find the point ( c ) in the interval ( 0 \leq x \leq 2 ) such that ( f(c) ) is equal to the average value of ( f(x) = x^{2/3} ), we first need to calculate the average value of the function over the interval.

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

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

In this case, ( a = 0 ) and ( b = 2 ). So, the average value of ( f(x) = x^{2/3} ) over the interval ( [0, 2] ) is:

[ \text{Average value} = \frac{1}{2 - 0} \int_{0}^{2} x^{2/3} , dx ]

We integrate this expression with respect to ( x ) over the interval ( [0, 2] ) and divide by the width of the interval, which is ( 2 - 0 = 2 ), to find the average value of the function.

Once we have the average value, we set ( f(c) ) equal to this average value and solve for ( c ). In other words, we find the value of ( c ) such that ( f(c) = \text{Average value} ) within the interval ( [0, 2] ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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