Title: More functions
Course: Demonstrating STACK (v4.4.7)

Q: Tick all properties that the function \(f(x)\),
\[f(x) = \cases{x, & when \(x \le -3\);\cr
-3, & when \(-3 < x < 4\);\cr
x-7,& when \(4 \le x\).\cr}\]
satisfies over the interval \([-5,-3]\).
- non-decreasing
- strictly increasing
- non-increasing
- strictly decreasing
- none of these
A: non-decreasing; strictly increasing

Find the equation of the oblique asymptote to the graph of the rational function

\[f(x)=\frac[value][value]\]

Enter your answer as a*[value] + b.

\(y = \)
Feedback: To find the equation of the oblique asymptote to the graph of the rational function \[f(x)=\frac[value][value]\] divide the numerator by the denominator. The equation is y = [value]

Find the [value]-value at which the function

 \[f(x)=[value],\]

has a point of inflexion (i.e. change in curvature).

The point of inflexion is at \(x = \)
Feedback: A point of inflexion occurs when \(\frac{d^2}{d[value]^2}\left([value]\right)\) = 0. \[\frac{d^2}{d[value]^2}\left([value]\right) = [value]\] Therefore the point of inflexion of the function \(f([value])\) occurs at \([value] = [value]\).

Find the [value]-value at which the function

\[f([value])=[value],\]

has its minimum.

The function has a minimum at \( x = \)
Feedback: To find the \([value]\)-value at which the function \(f([value])=[value]\), has turning points, differentiate and find the values of \([value]\) for which the differential is equal to 0. \[\begin{array}{rcl}\displaystyle \frac{d}{d[value]}\left([value]\right) &=& [value] \\ [value] & =& 0 \\ [value]&=& 0 \end{array}\] which factorises to give \[[value] = [value] \mbox{ and } [value] = -10\] A turning point is a minimum if \(\frac{d^2}{d[value]^2}\left([value]\right)\)>0. \[\frac{d^2}{d[value]^2}\left([value]\right) = [value]\] This is positive for \([value] = [value]\) and negative for \([value] = -10\). Therefore the minimum value of the function \(f([value])\) occurs at \([value] = [value]\).

Two positive numbers have product \([value]\). Using calculus, find the smallest possible value of their sum.

The smallest value of their sum is:
Feedback: The Product \[ xy = [value]\] \[y = \frac[value]{x}\] \[Sum = x + y = x +\frac[value]{x}= x + [value]^{-1}\] \[\frac{d(Sum)}{dx}= 1-[value]^{-2} \] To find the minimum \(\frac{d(Sum)}{dx}\) = 0. Therefore \[1-[value]^{-2} = 0\] \[x^{2} = [value]\] \[x = ±\sqrt[value]\] \(x\) and \(y\) must be positive hence the minimum sum is \[x + y = \sqrt[value]+\sqrt[value]=[value]\]

Given that \(\ln(a)=[value]\) , express: \[\log_{a}(x^[value])\] as a single multiple of a natural logarithm.

\(\log_{a}(x^[value]) = \)
Feedback: Below are the basic rules of logarithms. \[\log_{b}(mn) = \log_{b}(m) +\log_{b}(n)\] \[\log_{b}\left(\frac{m}{n}\right) = \log_{b}(m) -\log_{b}(n)\] \[\log_{b}(m^n) = n\cdot\log_{b}(m)\] \[\log_{a}(m)=\frac{\log_{b}(m)}{\log_{b}(a)}\] Hence \[\log_{a}(x^[value]) = [value]\log_{a}(x) = [value]\frac{\ln(x)}{\ln(a)}\] \[ = [value]\frac{\ln(x)}[value]=[value]\]

Find all real solutions, if any, of the following equation

\[ [value] \sinh(x) + [value] \cosh(x) - [value] = 0 .\]

(Do not use arcsinh or arccosh or arctanh in your answer. Express you answer in terms of a logarithm.)

\( x = \)
Feedback: If \[ [value] \sinh(x) + [value] \cosh(x) - [value] = 0 .\]then \[ [value] \sinh(x) + [value] \cosh(x) = [value] .\] \[ \sinh(x) + \cosh(x) = [value].\] Therefore\[ \frac{e^{x}-e^{-x}}{2} + \frac{e^{x}+e^{-x}}{2} = [value].\] \[ e^{x} = [value].\] and \[x =[value]\]