using mean value theorem

Question

justify y=e^x over [0,1]

Answer 1

not sure what you mean by justify, but since
y(0) = 1
y(1) = e
the slope of the line segment joining (0,1) and (1,e) is e-1,
we want c such that
y'(c) = e-1
e^x = e-1
x = ln(e-1) = 0.54 which is in the interval (0,1)