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Calculus, MET2 2024 VCAA 10 MC

Suppose a function  \(f:[0,5] \rightarrow R\)  and its derivative  \(f^{\prime}:[0,5] \rightarrow R\)  are defined and continuous on their domains. If  \(f^{\prime}(2)<0\)  and  \(f^{\prime}(4)>0\), which one of these statements must be true?

  1. \(f\) is strictly decreasing on \([0,2]\).
  2. \(f\) does not have an inverse function.
  3. \(f\) is positive on \([4,5]\).
  4. \(f\) has a local minimum at  \(x=3\).
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\(B\)

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\(\text{As}\ \ f^{\prime}(2)<0\ \ \text{and}\ \ f^{\prime}(4)>0\ \ \text{the function must have} \)

\(\text{a turning point between}\ \ x=2\ \ \text{and}\ \ x=4.\)

\(\therefore\ f(x)\ \text{will not have an inverse function, as it is many-to-one.}\)

\(\Rightarrow B\)

♦♦ Mean mark 38%.

Calculus, MET2 2020 VCAA 6 MC

Part of the graph of  `y=f^(')(x)`  is shown below.
 

 The corresponding part of the graph of  `y=f(x)`  is best represented by 
 

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`B`

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`text(By elimination:)`

`text(Two stationary points exist where)\ \ f^{‘}(x) =0\ \ text(and)\ \ x<0.`

`=>\ text(Eliminate)\ \ E, C, D`

`text{No other stationary points exist}\ →\ f^{‘}(x) !=0\ \ text(for)\ \ x>0.`

`=>\ text(Eliminate)\ \ E`

`=> B`