For what values of the parameter a does the function f (x) = x3 + 3(a − 7) x2 + 3(a2 − 9)x −1 have a positive
point of maximum?
Solution: f '(x) = 3x2 + 6(a − 7) x + 3(a2 − 9)
For f (x) to have a maximum at some point,
f '(x) = 0 and f "(x) < 0 for that point
Now, f '(x) = 0
⇒3x2 + 6(a − 7) x + 3(a2 − 9) = 0
⇒ x2 + 2(a − 7)x + (a2 − 9) = 0
⇒ x = −(a − 7)± 58 −14a ... (i)
For f'(x) to have real roots,
58 – 14a > 0
29
7
⇒ a < 29/7 (ii)
Now we determine which of the roots of f '(x) in (i) will give a local maximum and which will give a
local minimum.
f "(x) = 6x + 6(a − 7)
= 6(x + a − 7)
At x1 = − a − 7 + 58 −14a ⇒ f " x = 6 √58 −14a > 0
At x2 = − a − 7 − 58 −14a ⇒ f " x = −6 √58−14a < 0
so finally we need to use POSITIVE constraint.......now substituting x2 in original function will make it a NIGHTMARE!!!!!!!..how to find a easy way out then??
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1 Answers
1easy to make simple:
from graphs u know that if maxima occur at positive then minima will also occur at positive ((only for this given equation not true for all)).
so the roots of F'(x) must be positive and distinct
so a2 > 9
and a < 7
and also a < 29/7