Problem source

Let $f(x) = x^m(x-1)^n$, where $m$ and $n$ are both integers greater than $1$. Show that the curve $y=f(x)$ has a stationary point with $0 < x < 1$. By considering $f''(x)$, show that this stationary point is a maximum if $n$ is even and a minimum if $n$ is odd. Sketch the graphs of $f(x)$ in the four cases that arise according to the values of $m$ and $n$.

Solution source

\begin{align*}
&& f'(x) &= mx^{m-1}(x-1)^n + nx^m(x-1)^{n-1} \\
&&&= (m(x-1)+nx)x^{m-1}(x-1)^{n-1} \\
&&&= (x(m+n) - m)x^{m-1}(x-1)^{n-1} \\
\end{align*}

Therefore when $x = \frac{m}{m+n}$ there is a stationary point with $0 < x < 1$.

\begin{align*}
&& f''(x) &= m(m-1)x^{m-2}(x-1)^n + 2mnx^{m-1}(x-1)^{n-1} + n(n-1)x^{m}(x-1)^{n-2} \\
&&&= (m(m-1)(x-1)^2 +2mnx(x-1)+n(n-1)x^2)x^{m-2}(1-x)^{n-2} \\
\Rightarrow && f'' \left ( \frac{m}{m+n} \right) &= \left ( m(m-1) \frac{n^2}{(m+n)^2} - 2mn\frac{mn}{(m+n)^2} + n(n-1) \frac{m^2}{(m+n)^2} \right) \frac{m^{m-2}}{(m+n)^{m-2}} \frac{(-1)^{n-2}n^{n-2}}{(m+n)^{n-2}} \\
&&&= (-1)^{n-2}\frac{m^{m-1}n^{n-1}}{(m+n)^{m+n-2}} \left ( (m-1)n-2mn+(n-1)m\right) \\
&&&= (-1)^{n-2}\frac{m^{m-1}n^{n-1}}{(m+n)^{m+n-2}} \left ( -m-n\right) \\
&&&= (-1)^{n-1} \frac{m^{m-1}n^{n-1}}{(m+n)^{m+n-3}} 
\end{align*}

Therefore this is positive (and a minimum) when $n$ is odd and negative (and a maximum) when $n$ is even.


\begin{center}
    \begin{tikzpicture}
    \def\m{3};
    \def\n{4};
    \def\t{((\m)/(\m+\n))};
    \def\functionf(#1){((#1)^\m * ((#1)-1)^\n)/abs(\t^\m * (\t-1)^\n)};
    \def\xl{-0.2};
    \def\xu{1.2};
    \def\yl{-0.1};
    \def\yu{1.1};
    
    % Calculate scaling factors to make the plot square
    \pgfmathsetmacro{\xrange}{\xu-\xl}
    \pgfmathsetmacro{\yrange}{\yu-\yl}
    \pgfmathsetmacro{\xscale}{10/\xrange}
    \pgfmathsetmacro{\yscale}{10/\yrange}
    
    % Define the styles for the axes and grid
    \tikzset{
        axis/.style={very thick, ->},
        grid/.style={thin, gray!30},
        x=\xscale cm,
        y=\yscale cm
    }
    
    % Define the bounding region with clip
    \begin{scope}
        % You can modify these values to change your plotting region
        \clip (\xl,\yl) rectangle (\xu,\yu);
        
        % Draw a grid (optional)
        % \draw[grid] (-5,-3) grid (5,3);
        
        \draw[thick, blue, smooth, domain=\xl:\xu, samples=200] 
            plot (\x, {\functionf(\x)});
    \end{scope}
    
    % Set up axes
    \draw[axis] (\xl,0) -- (\xu,0) node[right] {$x$};
    \draw[axis] (0,\yl) -- (0,\yu) node[above] {$y$};

    \node[below] at (1,0) {$1$};

    \filldraw ({\t}, {\functionf(\t)}) circle (1pt) node[right] {$\left (\frac{m}{m+n}, f(\frac{m}{m+n}) \right)$};
    
    \end{tikzpicture}
\end{center}


\begin{center}
    \begin{tikzpicture}
    \def\m{4};
    \def\n{4};
    \def\t{((\m)/(\m+\n))};
    \def\functionf(#1){((#1)^\m * ((#1)-1)^\n)/abs(\t^\m * (\t-1)^\n)};
    \def\xl{-0.12};
    \def\xu{1.12};
    \def\yl{-0.1};
    \def\yu{1.1};
    
    % Calculate scaling factors to make the plot square
    \pgfmathsetmacro{\xrange}{\xu-\xl}
    \pgfmathsetmacro{\yrange}{\yu-\yl}
    \pgfmathsetmacro{\xscale}{10/\xrange}
    \pgfmathsetmacro{\yscale}{10/\yrange}
    
    % Define the styles for the axes and grid
    \tikzset{
        axis/.style={very thick, ->},
        grid/.style={thin, gray!30},
        x=\xscale cm,
        y=\yscale cm
    }
    
    % Define the bounding region with clip
    \begin{scope}
        % You can modify these values to change your plotting region
        \clip (\xl,\yl) rectangle (\xu,\yu);
        
        % Draw a grid (optional)
        % \draw[grid] (-5,-3) grid (5,3);
        
        \draw[thick, blue, smooth, domain=\xl:\xu, samples=200] 
            plot (\x, {\functionf(\x)});
    \end{scope}
    
    % Set up axes
    \draw[axis] (\xl,0) -- (\xu,0) node[right] {$x$};
    \draw[axis] (0,\yl) -- (0,\yu) node[above] {$y$};

    \node[below] at (1,0) {$1$};

    \filldraw ({\t}, {\functionf(\t)}) circle (1pt) node[right] {$\left (\frac{m}{m+n}, f(\frac{m}{m+n}) \right)$};
    
    \end{tikzpicture}
\end{center}



\begin{center}
    \begin{tikzpicture}
    \def\m{4};
    \def\n{3};
    \def\t{((\m)/(\m+\n))};
    \def\functionf(#1){((#1)^\m * ((#1)-1)^\n)/abs(\t^\m * (\t-1)^\n)};
    \def\xl{-0.12};
    \def\xu{1.12};
    \def\yl{-1.1};
    \def\yu{0.1};
    
    % Calculate scaling factors to make the plot square
    \pgfmathsetmacro{\xrange}{\xu-\xl}
    \pgfmathsetmacro{\yrange}{\yu-\yl}
    \pgfmathsetmacro{\xscale}{10/\xrange}
    \pgfmathsetmacro{\yscale}{10/\yrange}
    
    % Define the styles for the axes and grid
    \tikzset{
        axis/.style={very thick, ->},
        grid/.style={thin, gray!30},
        x=\xscale cm,
        y=\yscale cm
    }
    
    % Define the bounding region with clip
    \begin{scope}
        % You can modify these values to change your plotting region
        \clip (\xl,\yl) rectangle (\xu,\yu);
        
        % Draw a grid (optional)
        % \draw[grid] (-5,-3) grid (5,3);
        
        \draw[thick, blue, smooth, domain=\xl:\xu, samples=200] 
            plot (\x, {\functionf(\x)});
    \end{scope}
    
    % Set up axes
    \draw[axis] (\xl,0) -- (\xu,0) node[right] {$x$};
    \draw[axis] (0,\yl) -- (0,\yu) node[above] {$y$};

    \node[below] at (1,0) {$1$};

    \filldraw ({\t}, {\functionf(\t)}) circle (1pt) node[right] {$\left (\frac{m}{m+n}, f(\frac{m}{m+n}) \right)$};
    
    \end{tikzpicture}
\end{center}


\begin{center}
    \begin{tikzpicture}
    \def\m{3};
    \def\n{3};
    \def\t{((\m)/(\m+\n))};
    \def\functionf(#1){((#1)^\m * ((#1)-1)^\n)/abs(\t^\m * (\t-1)^\n)};
    \def\xl{-0.12};
    \def\xu{1.12};
    \def\yl{-1.1};
    \def\yu{0.1};
    
    % Calculate scaling factors to make the plot square
    \pgfmathsetmacro{\xrange}{\xu-\xl}
    \pgfmathsetmacro{\yrange}{\yu-\yl}
    \pgfmathsetmacro{\xscale}{10/\xrange}
    \pgfmathsetmacro{\yscale}{10/\yrange}
    
    % Define the styles for the axes and grid
    \tikzset{
        axis/.style={very thick, ->},
        grid/.style={thin, gray!30},
        x=\xscale cm,
        y=\yscale cm
    }
    
    % Define the bounding region with clip
    \begin{scope}
        % You can modify these values to change your plotting region
        \clip (\xl,\yl) rectangle (\xu,\yu);
        
        % Draw a grid (optional)
        % \draw[grid] (-5,-3) grid (5,3);
        
        \draw[thick, blue, smooth, domain=\xl:\xu, samples=200] 
            plot (\x, {\functionf(\x)});
    \end{scope}
    
    % Set up axes
    \draw[axis] (\xl,0) -- (\xu,0) node[right] {$x$};
    \draw[axis] (0,\yl) -- (0,\yu) node[above] {$y$};

    \node[below] at (1,0) {$1$};

    \filldraw ({\t}, {\functionf(\t)}) circle (1pt) node[right] {$\left (\frac{m}{m+n}, f(\frac{m}{m+n}) \right)$};
    
    \end{tikzpicture}
\end{center}