Discussion questions · Section 1 of 3

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.

The diagram shows $y=f^{\prime}(x)$, passing through the origin with turning points $(-3,0)$ and $(-1.5,-1)$. The x-axis and $x=1$ are its two asymptotes.

(i) Find the range of $x$ for which $y=f(x)$ is strictly increasing and concave downwards.

(ii) State the x-coordinates and natures of all stationary points of $y=f(x)$.

(iii) Given $f(0)=1$, sketch $y=f(x)$ for $x<1$, clearly indicating all stationary points, asymptotes and intersections with the axes.