Let h be a differentiable function on R and satisfy h0 1 h1

Let h be a differentiable function on R and satisfy h(0) = 1. h(1) = 2. h(2) = 1 Show that there exists a point d epsilon [0, 2] where h(d) = d Show that there exist two points a, b in (0.2) such that h(a) = 1. For every number k with |k|

Solution

It is given that h is a differentiable function on R [0,2] where h(0)=1,h(1)=2,h(2)=1

function h is increasing over [0,1] while decreasing over [1,2]

so when we moves from 1 to 2 ,value of h goes from 2 to 1,so definitely there is a point occurs when h(d)=d .

b) the value of h goes from 1 to 2 and then from 2 to 1 ,at h(1) curves changes from increasing to decreasing means 0n 1^- slope will be +1 and on 1⁺ slope will be -1 .This occurs only one times in the range R[0,2]

c)

from 0 to 1 slope will les than 1 and from 1 to 2 slope will be more than 1 ,or mode(slope)=less than 1 (k)