Use the Mean Value Theorem to show that the function fx arc

Solution

we have f(x)= tan^-1x

let a, b are some positive natural number corresponding to value of x such that its slope of f(x) =3

so by mean value theorem there must be a poibt between a and b such that,

3= (tan^-1b- tan^-1a)/(b-a) ...... equation A

now we know that

tan^-1x < x ..for all x>0

and tan^-1x > x for all x<0 .........its a formula one should really need to know

if we take positive values

so we have

tan^-1a < a ........... equation 1

and tan^-1b < b ...........equation 2

subtracting equation 1 from equation 2 we get:

(tan^-1b- tan^-1a) < (b-a)

or

(tan^-1b- tan^-1a)/(b-a) < 1 .........equation 3

as equation A and equation 3 cant be true at the same time, thus  we are in contradiction with equation A.

so its proven that the function f(x) = arctan(x)
does not have a tangent line with slope 3 at any point.