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Verify Lagranges mean value theorem f...

Verify Lagranges mean value theorem for function `f(x)=x^3-2x^2-x+3` on `[0,\ 1]` and find a point `' c '` in the indicated interval:

Text Solution

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`f(x)=x^{3}-2 x^{2}-x+3=0`

Since a polynomial function is everywhere continuous and differentiable.

Therefore,

`f(x)` is continuous on `[0,1]` and differentiable on `(0,1)`

Thus, both conditions of Lagrange's mean value theorem are satisfied.

So, there must exist at least one real number `c in(0,1)` such that `f^{prime}(c)=frac{f(1)-f(0)}{1-0}`

Now, `f(x)=x^{3}-2 x^{2}-x+3=0`

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