The graphs of $y=f'(x)$ and $y=|f(x)|$ are shown below. State the nature of all turning point(s)

Question

The graphs of $y=f'(x)$ and $y=|f(x)|$ are shown below. State the nature of all turning point(s) of the graph of $y=\mathrm{f}(x)$.[2] State the range of values of $x$ where $\mathrm{f}$ is decreasing.[2] Sketch the graph of $y=\mathrm{f}(x)$ indicating clearly the equations of the asymptotes, coordinates of the turning point(s) and the intersections with the axes.[3] On the copy of the graph of $y=|\mathrm{f}(x)|$ in the Printed Answer Booklet, sketch and label a line $y=kx+3k$, where $k$ is a constant. Hence state the range of values of $k$ for which there is no real solution to the equation $|\mathrm{f}(x)|=kx+3k$.[2]

Accepted Answer

(a) Maximum $(-2,-2)$; minimum $(1.5,-1)$ (b) $-2\le x<0$ or $0<x\le1.5$ (c) Asymptotes $x=0$, $y=x+3$, and $y=3$; turning points $(-2,-2)$ and $(1.5,-1)$; intercepts $(1,0)$ and $(2.5,0)$, with no $y$-intercept (d) $-1\le k<0$

2025 DHS P1 Q5

2025 DHS P1 Q5

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