Mathematics

Problem

Four smooth curves

W

X

Y

, and

Z

are defined for

x>0

W(x)=\sqrt{x}

X(x)=\frac{1}{2\sqrt{x}}

Y(x)=-\frac{1}{4}x^{-3/2}

Z(x)=\ln x

. Exactly three of these curves constitute the set {

f

f'

f''

} for some twice-differentiable function

f

. Determine which three curves belong to that set and label each as

f

f'

, or

f''

Intuitive explanation

The key is to recognize that if three curves form a function and its first two derivatives, they must satisfy a chain of differentiation: each curve should be the derivative of the previous one. Start by computing the derivatives of each given function and look for a closed loop—where differentiating one curve gives another curve in the set. Once you find three curves that chain together this way, you've identified your $f$ , $f'$ , and $f''$ . The fourth curve will be the outsider that doesn't fit the pattern.

Intuitive explanation

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