Practice section 4 of 8

How do we construct a cylindrical metal can with a fixed volume $V$ so that its surface area, the amount of metal used, is minimised?

The diagrams below show the graph of $f$ and its derivative $f^{\prime}$. State clearly, with reasons, which is the graph of $f$ and that of $f^{\prime}$.

The diagram shows the graph of $y=f(x)$. It has a maximum point $(4a,3a)$ and asymptotes $x=2a$ and $y=a$. The drawing also shows the curve passing through the origin and $(3a,0)$.

Sketch the graph of $y=f^{\prime}(x)$, where $f^{\prime}$ is the gradient function of $f$, showing the relevant features of the graph.

The diagram shows the graph of $y=g(x)$. Its intersections with the axes are $(0,1)$, $(1,0)$ and $(3,0)$. The asymptotes are $x=2$ and $y=-x+2$.

Sketch the graph of $y=g^{\prime}(x)$, making clear the main relevant features.

The diagram shows the graph of $y=|h(x)|$. Given that $h^{\prime}(x)<0$ for $x\in\mathbb{R}$, $x\ne0$, sketch the graph of $y=h(x)$.

The supplied modulus graph has vertical asymptote $x=0$, horizontal asymptote $y=a$, and a zero at $x=b>0$.

The diagram shows $y=f^{\prime}(x)$. The curve cuts the x-axis at $(a,0)$ and $(-a,0)$, has a stationary point at $(0,b)$, and has asymptote $y=k$.

(i) State the x-coordinates of the stationary points of $y=f(x)$ and determine the nature of each.

(ii) State the ranges of $x$ for which $y=f(x)$ is (a) strictly increasing, (b) concave downwards.