Functions with prescribed properties (2)

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Write a function (a different one for each item) that:

• (a) has a relative minimum at $x=0$ but is not differentiable there;
• (b) has an oblique asymptote of slope $m=-1$;
• (c) has a vertical asymptote at $x=-1$ and a horizontal asymptote at $y=0$;
• (d) has an inflection point at $x=4$ where the function also has zero derivative.

Solution

A construction exercise: one valid example is required for each property.

(a) The absolute value has a minimum with a corner at the origin: $\boxed{y=|x|}$ It has a relative (and absolute) minimum $0$ at $x=0$; there the right ($+1$) and left ($-1$) derivatives differ, so it is not differentiable at $x=0$.

(b) Slope $-1$ means an asymptote $y=-x+q$; just add to $-x$ an infinitesimal term: $\boxed{y=-x+\dfrac{1}{x}}$ As $x\to\pm\infty$ we get $y-(-x)=\dfrac1x\to0$, so $y=-x$ is an oblique asymptote of slope $-1$.

(c) A shifted hyperbola with pole at $x=-1$ and limit $0$ at infinity: $\boxed{y=\dfrac{1}{x+1}}$ As $x\to-1$ we get $y\to\pm\infty$: vertical asymptote $x=-1$. As $x\to\pm\infty$ we get $y\to0$: horizontal asymptote $y=0$.

(d) A shifted cubic: inflection with horizontal tangent at $x=4$: $\boxed{y=(x-4)^{3}}$ Indeed $y'=3(x-4)^2$ vanishes at $x=4$ (zero derivative) and $y''=6(x-4)$ changes sign at $x=4$: there is an inflection with a horizontal tangent.