Maximum of the derivative

Calculus Level 2

What is the supremum of the fourth derivative of f ( x ) = e a x f(x) = e^{ax} on the interval [ a , 0 ] [a, 0] for − 1 < a < 0 ? -1<a<0?

a 4 e a 2 a^4e^{a^2} 1 1 e a 2 e^{a^2} a 4 a^4

July Thomas by July Thomas , 30, USA

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1 solution

July Thomas
Jun 9, 2016

The fourth derivative is

f ( 4 ) ( x ) = a 4 e a x . f^{(4)}(x) = a^4 e^{ax}.

Additionally,

sup ⁡ [ e a x ] a 0 = e a 2 \sup[e^{ax}]_a^0 = e^{a^2} for − 1 < a < 0. -1<a<0.

So the supremum of the fourth derivative is just a 4 e a 2 . a^4 e^{a^2}.

15 Helpful 0 Interesting 1 Brilliant 21 Confused

For -1 < a < 0, isn't e^(a*a) > e^0?

Siva Bathula - 4 years, 1 month ago

very difficult example

Ahmadjon Kurbanov - 3 years, 7 months ago

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