How do you determine all values of c that satisfy the conclusion of the mean value theorem on the interval [0,1] for #f(x) = (x)arcsin(x)#?

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Answer 1 · 2017-01-31T22:21:42

You'll need some kind of numerical or technological method of solving #x/sqrt(1-x^2) + arcsinx = pi/2#

The #c# mentioned in the conclusion of MVT must satisfy:

#f'(x) = (f(1)-f(0))/(1-0)# which is the same as

#f'(x) = (arcsin(1) - 0)/1 = pi/2#.

Since #f'(x) = x/sqrt(1-x^2) + arcsinx #, finding #c# amounts to solving

#x/sqrt(1-x^2) + arcsinx = pi/2#

#c = x ~~ 0.652#