Which of the following functions is concave down over the set of positive real numbers:
A
f(x)=2x
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B
f(x)=x12
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C
f(x)=3x+2
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D
f(x)=x3
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Solution
The correct option is Bf(x)=x12 A function f(x) is concave down if f′′(x)<0.
Now, checking options: (A)f′′(x)=(ln2)22x (B)f′′(x)=−14√x3 (C)f′′(x)=0 (D)f′′(x)=6x
Only (B) has a negative second derivative for all positive real numbers.