Let be the function defined for by f(x)=a ,x^ -2 +b ,x^ 1/2

MATHEMATICS · Multi-part · A Level · Mathematics · Differentiation · Differentiation

Problem

Let

f

be the function defined for

x>0

f(x)=a\,x^{-2}+b\,x^{1/2},

where

a

and

b

are non-zero real constants. (a) Find

f'(x)

in terms of

a

and

b

. (b) If

a=12

and

b=-3

, determine the slope of the tangent line to the graph of

f

x=4

. (c) Using the values of

a

and

b

from part (b), write the equation of the tangent line to the graph of

f

x=4

in point-slope form. (d) For general constants

a

and

b

, determine all

x>0

at which the tangent line to the graph of

f

is horizontal. Express your answer in terms of

a

and

b

and justify that the tangent line is horizontal precisely at those points.