Question

It is given that the Rolle's theorem holds for the function f(x) = x³ + bx² + cx, x $\in$ [1, 2] at the point x = $\frac{4}{3}$. Find the values of b and c.

Answer 1

As, the Rolle's theorem holds for the function f(x) = x³ + bx² + cx, x $\in$ [1, 2] at the point x = $\frac{4}{3}$

$$\begin{gathered} \mathrm{So},f\left(1\right)=f\left(2\right) \\ \Rightarrow {\left(1\right)}^3+b{\left(1\right)}^2+c\left(1\right)={\left(2\right)}^3+b{\left(2\right)}^2+c\left(2\right) \\ \Rightarrow 1+b+c=8+4b+2c \\ \Rightarrow 3b+c+7=0.....\left(\mathrm{i}\right) \\ \mathrm{And}f\text{'}\left(\frac{4}{3}\right)=0 \\ \Rightarrow 3{\left(\frac{4}{3}\right)}^2+2b\left(\frac{4}{3}\right)+c=0\left[\mathrm{As},f\text{'}\left(x\right)=3x^2+2bx+c\right] \\ \Rightarrow \frac{16}{3}+\frac{8b}{3}+c=0 \\ \Rightarrow 8b+3c+16=0.....\left(\mathrm{ii}\right) \\ \left(\mathrm{ii}\right)-\left(\mathrm{i}\right)\times 3,\mathrm{we}\mathrm{ge} \\ 8b-9b+16-21=0 \\ \Rightarrow -b-5=0 \\ \Rightarrow b=-5 \\ \mathrm{Substituting}b=-5\mathrm{in}\left(\mathrm{i}\right),\mathrm{we}\mathrm{get} \\ 3\left(-5\right)+c+7=0 \\ \Rightarrow -15+c+7=0 \\ \Rightarrow c=8 \end{gathered}$$