How do I find the numbers #c# that satisfy Rolle's Theorem for #f(x)=sqrt(x)-x/3# on the interval #[0,9]# ?

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Answer 1 · 2014-09-22T03:20:38

#c=9/4#

Let us look at some details.

By taking the derivative,

#f'(x)=1/{2sqrt{x}}-1/3#

By solving #f'(c)=0#,

#f'(c)=1/{2sqrt{c}}-1/3=0#

by adding #1/3#,

#Rightarrow 1/{2sqrt{c}}=1/3#

by taking the reciprocal,

#Rightarrow 2sqrt{c}=3#

by dividing by 2,

#Rightarrow sqrt{c}=3/2#

by squaring,

#Rightarrow c=9/4#