Mean Value Theorem

Verify that the function $f(x)= 2x^3+3x$ satisfies the hypothesis of the mean value theorem on the interval [0,2]. Find all the numbers C that satisfy the conclusion of the Mean Value Theorem. 

The mean value of the first derivative of the function on the interval [0,2] is
$[f(2)-f(0)]/(2-0) =(2*2^3+3*2 -0)/2 =(16+6)/2 =11$
the first derivative of the function is $f'(x) =6x^2+3$
Now one must have $f'(x) =11$
$6x^2+3=11$
$6x^2 =8$
$x^2 =4/3$
$x1 =2/\sqrt{(3)}$
$x2 =-2/\sqrt{(3)}$
X1 and x2 are the numbers C that you satisfy the conclusion of the mean value theorem


valentin68

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