Let , where and are real parameters. (a) Show that is a critical point of if and only…

MATHEMATICS · Subjective · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Let

f(x)=x^{3}+ax^{2}+bx+1

, where

a

and

b

are real parameters. (a) Show that

x=1

is a critical point of

f

if and only if

3+2a+b=0

. (b) Under the condition found in part (a), derive the additional inequality involving

a

and

b

that ensures

f'(x)<0

for all

0<x<1

and

f'(x)>0

for all

x>1

. (c) Hence determine the set of ordered pairs

(a,b)

for which

f

is strictly decreasing on

(0,1)

and strictly increasing on

(1,\infty)