Show stationary point exists or gradient has specific property

Use differentiation to show that a stationary point exists at a given location, or that the derivative always satisfies some inequality (e.g., always negative, always positive).

7 questions · Moderate -0.3

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CAIE P2 2010 June Q5

7 marks Moderate -0.8

5 The equation of a curve is $y = x ^ { 3 } \mathrm { e } ^ { - x }$.

Show that the curve has a stationary point where $x = 3$.
Find the equation of the tangent to the curve at the point where $x = 1$.

OCR MEI C3 2008 June Q5

6 marks Moderate -0.3

5 Show that the curve $y = x ^ { 2 } \ln x$ has a stationary point when $x = \frac { 1 } { \sqrt { \mathrm { e } } }$.

OCR C3 Q8

12 marks Standard +0.3

A curve has the equation $y = \frac { \mathrm { e } ^ { 2 } } { x } + \mathrm { e } ^ { x } , x \neq 0$.
1. Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
  [0pt]
2. Show that the curve has a stationary point in the interval [1.3,1.4].
The point $A$ on the curve has $x$-coordinate 2 .
Show that the tangent to the curve at $A$ passes through the origin. The tangent to the curve at $A$ intersects the curve again at the point $B$.
The $x$-coordinate of $B$ is to be estimated using the iterative formula $$x _ { n + 1 } = - \frac { 2 } { 3 } \sqrt { 3 + 3 x _ { n } \mathrm { e } ^ { x _ { n } - 2 } }$$ with $x _ { 0 } = - 1$.
Find $x _ { 1 } , x _ { 2 }$ and $x _ { 3 }$ to 7 significant figures and hence state the $x$-coordinate of $B$ to 5 significant figures.

OCR C3 2011 January Q9

13 marks Standard +0.3

The function f is defined for all real values of $x$ by $$f ( x ) = e ^ { 2 x } - 3 e ^ { - 2 x } .$$
1. Show that $\mathrm { f } ^ { \prime } ( x ) > 0$ for all $x$.
2. Show that the set of values of $x$ for which $\mathrm { f } ^ { \prime \prime } ( x ) > 0$ is the same as the set of values of $x$ for which $\mathrm { f } ( x ) > 0$, and state what this set of values is.
3. \includegraphics[max width=\textwidth, alt={}, center]{774bb427-5392-45d3-8e4e-47d08fb8a792-04_634_830_641_699} The function g is defined for all real values of $x$ by $$\mathrm { g } ( x ) = \mathrm { e } ^ { 2 x } + k \mathrm { e } ^ { - 2 x } ,$$ where $k$ is a constant greater than 1 . The graph of $y = \mathrm { g } ( x )$ is shown above. Find the range of g , giving your answer in simplified form.

OCR MEI Paper 3 2020 November Q11

2 marks Easy -2.5

11 Show that $\mathrm { e } ^ { x }$ is an increasing function for all values of $x$, as stated in line 39 .

Edexcel C3 Q8

13 marks Standard +0.3

8. A curve has the equation $y = \frac { \mathrm { e } ^ { 2 } } { x } + \mathrm { e } ^ { x } , \quad x \neq 0$.

Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
[0pt]
Show that the curve has a stationary point in the interval [1.3,1.4]. The point $A$ on the curve has $x$-coordinate 2 .
Show that the tangent to the curve at $A$ passes through the origin. The tangent to the curve at $A$ intersects the curve again at the point $B$.
The $x$-coordinate of $B$ is to be estimated using the iterative formula $$x _ { n + 1 } = - \frac { 2 } { 3 } \sqrt { 3 + 3 x _ { n } \mathrm { e } ^ { x _ { n } - 2 } }$$ with $x _ { 0 } = - 1$.
Find $x _ { 1 } , x _ { 2 }$ and $x _ { 3 }$ to 7 significant figures and hence state the $x$-coordinate of $B$ to 5 significant figures.

Edexcel CP2 2021 June Q5

8 marks Standard +0.8

The curve $C$ has equation

$$y = \arccos \left( \frac { 1 } { 2 } x \right) \quad - 2 \leqslant x \leqslant 2$$

Show that $C$ has no stationary points. The normal to $C$, at the point where $x = 1$, crosses the $x$-axis at the point $A$ and crosses the $y$-axis at the point $B$. Given that $O$ is the origin,
show that the area of the triangle $O A B$ is $\frac { 1 } { 54 } \left( p \sqrt { 3 } + q \pi + r \sqrt { 3 } \pi ^ { 2 } \right)$ where $p$, $q$ and $r$ are integers to be determined.
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