linear interpolation error bound

I am to show:

|u'(x) - p'(x)| <= h max|u''(y)| y in (0,1).

where p(x) is a linear polynomial.

However, I am actually getting h/2 max|u''(y)| y in (0,1). Here is my work:

By the polynomial interpolation error, for x in [jh, (j+1)h]:

u'(x) - p'(x) = u''(y(x))/2 d/dx[(x - jh)(x - (j+1)h)]

|u'(x) - p'(x) |<= max|u''(y)| / 2 y in (0,1) * (2x - 2jh -h)

|u'(x) - p'(x) |<= max|u''(y)| y in (0,1) * (x - jh -h/2)

Taking x = (j+1)h I get:

|u'(x) - p'(x) |<= h/2 * max|u''(y)| y in (0,1)


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