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Calculus, 2ADV C3 2025 HSC 10 MC

The graph of  \(y=f(x)\), with all its stationary points, is shown.
 

How many stationary points does the graph of  \(y=f\left(e^x\right)\)  have?

  1. 0
  2. 1
  3. 2
  4. 3
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\(C\)

Show Worked Solution

\(y=f(e^{x})\ \ \Rightarrow\ \ y^{\prime}=e^{x} \times f(e^{x}) \)

\(\text{Find number of \(x\) values where}\ \ y^{\prime}=0.\)

\(\text{Since}\ e^{x} \in (0, \infty)\ \text{for all}\ x: \)

\(\text{Stationary points of \(f(e^x)\) = 2 (SP’s of \(f(x)\) for}\ x \in (0, \infty)).\)

\(\Rightarrow C\)

Calculus, 2ADV C3 2020 HSC 10 MC

The graph shows two functions  `y = f(x)`  and  `y = g(x)`.

Define  `h(x) = f(g(x))`.

How many stationary points does  `y = h(x)`  have for  `1 <= x <= 5`?

  1. 0
  2. 1
  3. 2
  4. 3
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`D`

Show Worked Solution

`h(x) = f(g(x))`

♦♦♦ Mean mark 11%.

`h′(x) = g′(x) xx f′(g(x))`
  

`text(S.P.’s occur when)\ \ g′(x) = 0\ \ text(or)\ \ f′(g(x)) = 0`

`g′(x) = 0\ \ text(when)\ \ x =3\ (text(from graph))`

`f′(x) = 0\ \ text(when)\ \ x ~~ 1  \ \ text{(i.e.}\ xtext{-value is just under 1)}`
 

`text(Find values of)\ x\ text(when)\ g(x) ~~ 1:`

`text(By inspection, there are 2 values where)`

`g(x) ~~ 1, \ x ∈ [1, 5]`

`:.\ text(There are 3 S.P.’s for)\ y = h(x), \ x ∈ [1, 5]`

`=> D`