Maxima and MinimaMedium
Question
For all a, b ∈ R the function f(x) = 3x4 - 4x3 + 6x2 + ax + b has -
Options
A.no extremum
B.exactly one extremum
C.exactly two extremum
D.three extremum
Solution
f(x) = 3x4 - 4x3 + 6x2 + ax + b
f′(x) = g(x) =12x3 - 12x2 +12x + a
f″(x) = 36x2 - 24x +12
= 12(3x2 - 2x + 1)
f″(x) > 0
f″(x) is increasing
⇒ f′(x) = 0 at exactly one point.
⇒ The given function has exactly one extremum.
f′(x) = g(x) =12x3 - 12x2 +12x + a
f″(x) = 36x2 - 24x +12
= 12(3x2 - 2x + 1)
f″(x) > 0
f″(x) is increasing
⇒ f′(x) = 0 at exactly one point.
⇒ The given function has exactly one extremum.
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