A function is defined by for all real numbers , where and are positive integers. What is the local maximum of this function?
Notation: denotes the absolute value function .
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Note that f ( x ) ≥ 0 for all x , f ( − ∞ ) = f ( ∞ ) = ∞ and f ( 0 ) = f ( 1 ) = 0 . Therefore, the local maximum must be within x ∈ ( 0 , 1 ) . Then we have:
f ( x ) f ′ ( x ) m x m − 1 ( 1 − x ) n m ( 1 − x ) ⟹ x ⟹ max ( f ( x ) ) = ∣ x ∣ m ⋅ ∣ x − 1 ∣ n = x m ( 1 − x ) n = m x m − 1 ( 1 − x ) n − n x m ( 1 − x ) n − 1 = n x m ( 1 − x ) n − 1 = n x = m + n m = f ( m + n m ) = ( m + n m ) m ( 1 − m + n m ) n = ( m + n ) m + n m m n n For x ∈ ( 0 , 1 ) Differentiate both sides w.r.t. x Putting f ′ ( x ) = 0 For x = 0