Verify Lagranges mean value theorem for function `f(x)=x^2-2x+4` on `[1,\ 5]` and find a point `' c '` in the indicated interval:

Answer

Proved

Explanation

ATQ,
$$ f(x)=x^{2}-2 x+4 $$
As we know polynomial functions are continuous and differentiable at each point. Therefore, `f(x)` is continuous on `[1,5]` and differentiable on `(1,5)`.

Thus, both conditions of LMVT are satisfied.

So, there must exist one real number


`c \in(1,5)` such that `f^{\prime}(c)=\frac{f(5)-f(1)}{5-1}=\frac{f(5)-f(1)}{4}`

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