Question

State when a function f(x) is said to be increasing on an interval [a, b]. Test whether the function f(x) = x² − 6x + 3 is increasing on the interval [4, 6].

Answer 1

$$\begin{gathered} \mathrm{A}\mathrm{function}f\left(x\right)\mathrm{is}\mathrm{said}\mathrm{to}\mathrm{be}\mathrm{increasing}\mathrm{on}\left[a,b\right]\mathrm{if}\mathrm{it}\mathrm{is}\mathrm{increasing}\mathrm{at}x=a\mathrm{and}x=b. \\ \mathrm{Here}, \\ f\left(x\right)=x^2-6x+3 \\ f\text{'}\left(x\right)=2x-6 \\ \Rightarrow f\text{'}\left(x\right)=2\left(x-3\right) \\ \mathrm{Now},f\text{'}\left(4\right)=2\left(4-3\right) \\ =2 \\ \therefore f\text{'}\left(4\right)>0 \\ \mathrm{So},f\left(x\right)\mathrm{is}\mathrm{increasing}\mathrm{on}x=4 \\ \&,f\text{'}\left(6\right)=2\left(6-3\right) \\ =6 \\ \therefore f\text{'}\left(6\right)>0 \\ \mathrm{So},f\left(x\right)\mathrm{is}\mathrm{increasing}\mathrm{on}x=6 \\ \text{Hence,}\mathit{\text{f}}\left(x\right)\text{is increasing on [4, 6].} \end{gathered}$$