Q.Verify Lagrange's mean value theorem for the function $f(x) = 2x - x^2,\ x \in [0, 1]$.

Lagrange's Mean Value Theorem (LMVT) generalises Rolle's Theorem to the case where $f(a)$ need not equal $f(b)$. It states that if $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, then there exists at least one $c$ in $(a,b)$ such that $f'(c)=\dfrac{f(b)-f(a)}{b-a}$. Geometrically, the right-hand side is the slope of the chord (secant line) joining the endpoints $A(a,f(a))$ and $B(b,f(b))$ on the curve; the theorem guarantees that somewhere between $A$ and $B$ the tangent to the curve is exactly parallel to this chord. To verify LMVT for a given function: confirm continuity/differentiability on the given interval, compute $f(a)$ and $f(b)$ and hence t …