A function $\mathrm{f}$ has the equation \(\mathrm{f}\left( x \right)=\frac{1}{5}{{x}^{5}}+2{{x}^{

Question

A function $\mathrm{f}$ has the equation $\mathrm{f}\left( x \right)=\frac{1}{5}{{x}^{5}}+2{{x}^{3}}+10x-25$. Without using a calculator, and by considering $\mathrm{f}'\left( x \right)$, explain why the equation $\mathrm{f}\left( x \right)=0$ has only one real root between $0$ and $2$.[2] Find the root, giving your answer to $3$ decimal places.[1] Using a non-graphical method, find the range of values of $x$ for which $\mathrm{f}$ is concave downward.[2] By considering a sketch of $y=\mathrm{f}\left( x \right)$ and another suitable curve on the same diagram, find the range of values of $k$ such that the equation $\frac{1}{5}{{x}^{5}}+2{{x}^{3}}+\left( 10+k \right)x=0$ has $3$ distinct real roots.[3]

Accepted Answer

(a) ${\mathrm{f}}$ is an increasing function. Thus, ${\mathrm{f}(x)=0}$ has at most one real root. (b) $x=1.559$ (c) $x<0$ (d) $k<-10$

2022 DHS Promo Q8

2022 DHS Promo Q8

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