Is Lagrenge's mean value theorem applicable to f(x)=|x| on the interval [-1,2]
No.
check f(0)
@steve... can u please elaborate ?
To determine if Lagrange's mean value theorem is applicable to the function f(x) = |x| on the interval [-1, 2], we need to check two conditions:
1. Continuity: The function f(x) = |x| is continuous on the interval [-1, 2]. A function is continuous if it has no holes, jumps, or vertical asymptotes within the interval. In this case, the absolute value function |x| is continuous everywhere, so it is continuous on the interval [-1, 2].
2. Differentiability: The function f(x) = |x| needs to be differentiable on the interval (-1, 2). However, the absolute value function |x| is not differentiable at x = 0 because it has a sharp corner (cusp) at that point. Therefore, Lagrange's mean value theorem is not applicable to f(x) = |x| on the interval [-1, 2].
In summary, since the function f(x) = |x| is not differentiable at x = 0, Lagrange's mean value theorem cannot be applied to this function on the interval [-1, 2].