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Calculus, MET2 2025 VCAA 16 MC

Consider the function  \(h(x)=a\, \log _e(b x)\), where \(a, b \in R \backslash\{0\}\).

Given that its derivative \(h^{\prime}(x)\) has range \((0, \infty)\), which of the following must be true?

  1. \( a>0\)  only
  2. \( a>0\)  and  \(b<0\)
  3. \(a>0\)  and  \(b>0\)
  4. \(a b>0\)
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\(D\)

Show Worked Solution

\(h(x)=a\, \log _e(b x)\)

\(h^{\prime}(x)=a\, \times \dfrac{b}{bx}=\dfrac{a}{x}\)

\(\text{Consider} \ \ h(x)=a\, \log _e(b x) \ \Rightarrow \ b x>0\)

♦♦♦ Mean mark 18%.

\(\text{Case 1:} \ \ b>0 \ \Rightarrow \ x>0 \ (\text{since} \ \ b x>0 )\)

\(\dfrac{a}{x} \ \ \text{is only positive when}\ \  a>0\)
 

\(\text {Case 2:} \ \ b<0 \ \Rightarrow \ x<0 \ \ (\text{since} \ \ b x>0)\)

\(\dfrac{a}{x} \ \ \text{is only positive when} \ \  a<0\)

\(\text{In both cases,} \ ab>0\)

\(\Rightarrow D\)