Question

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?

Answer 1

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]`