Given the function # f(x)=3x^3−2x#, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [-4,4] and find the c?

1 Answer
Apr 1, 2017

The values of #c# are #=+-2.31#

Explanation:

The mean value states theorem that if a function is continous on an

interval #[a,b]# and derivable on the interval #]a,b[#, then there is a

point #c in [a,b] # such that #f'(c)=(f(b)-f(a))/(b-a)#

Here,

#f(x)=3x^3-2x# is a polynomial function, defined, continuous and derivable on the interval #x in [-4,4]#

#f'(x)=9x^2-2#

#f(-4)=3*(-4)^3+8=-192+8=-184#

#f(4)=3*4^3-8=184#

Therefore,

#f'(c)=(f(4)-f(-4))/(4+4)=(184+184)/8=46#

#f'(c)=9c^2-2=46#

#c^2=(46+2)/9=sqrt48/3#

#c=+-4sqrt3/3=+-2.31#

Therefore,

#c in [-4,4]#