`f(x) = x^4 - 8x, [0,2]` Determine whether the Mean Value Theorem can be applied to `f` on the closed interval `[a,b]`. If the Mean Value Theorem can be applied, find all values of `c` in the open interval `(a,b)` such that `f'(c) = (f(b) - f(a))/(b - a)`. If the Mean Value Theorem cannot be applied, explain why not.
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The mean value theorem may be applied to the given function since all polynomial functions are continuous and differentiable on R, hence, the given function is continuous on [0,2] and differentiable on (0,2).
If the function is continuous and differentiable over the given interval, then, there exists a point `c in (0,2), ` such that:
`f(2) - f(0) = f'(c)(2 - 0)`
You need to evaluate f(2) and f(0):
`f(2) = 2^4 - 8*2 => f(2) = 16 - 16 = 0`
`f(0) = 0^4 - 8*0 = 0`
You need to evaluate `f'(c) = (c^4 - 8c)' => f'(c) = 4c^3 - 8`
Replace the found values in equation f(2) - f(0) = f'(c)(2 - 0):
`0 - 0 = 2(4c^3 - 8) => 4c^3 - 8 = 0 => 4c^3= 8 => c^3 = 8/4 => c^3 = 2 => c = root(3)(2)`
Hence, evaluating if the mean value theorem is applicable, yields that it is. The value of `c in (0,2)` is `c = root(3)(2).`
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