For what real values of ‘a’ and ‘b’ all the extrema of the function f(x) = 
x 3 + 2ax 2 – 9x + b
are positive and the maximum is at the point x 0 = 
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
(If a = 
, then b > 
; If a = 
then b > 
)
Since we have a cubic polynomial with positive leading coefficient the maxima will occur before minima.
f’(x) = 5a 2 x 2 + 4ax – 9
put f’(x) = 0 ⇒ x = 
, 
Case - 1
a < 0
then maxima occurs when x = 
⇒ 
= – 
⇒ a = – 
Minima therefore occurs when x = 
= 1
⇒ f(1) > 0 when a = – 
⇒ 
+ 2a – 9 + b > 0
– 
– 9 + b > 0
> 0 ⇒ 9 – 
⇒ b > 
Case - 2 a > 0
f 
> 0 ⇒ 
+ b > 0
2 
– 9. 
+ b > 0
. 
+ 2. 
– 9 
+ b > 0
b + 
= 0
b + 
(+16) ⇒ b > 
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