Question

If f(x) = x³ + ax² + bx + c has a maximum at x= $-$1 and minimum at x=3. Determine a,b and c.

Answer 1

$$\begin{gathered} \mathrm{Given}:\hspace{0.17em}y=x^3+a{x}^2+bx+c \\ \Rightarrow \frac{dy}{dx}=3x^2+2ax+b \\ \mathrm{S}\text{ince}\mathit{\text{f}}\left(x\right)\text{is at maximum at}\mathit{\text{x}}\text{=-1 and minimum at}\mathit{\text{x}}\text{=3},x=-1\mathrm{and}x=3\mathrm{are}\mathrm{the}\mathrm{critical}\mathrm{points}. \\ \mathrm{At}x=-1: \\ \frac{dy}{dx}=3{\left(-1\right)}^2+2a\left(-1\right)+b=0 \\ \Rightarrow 3-2a+b=0 \\ \Rightarrow 2a-b=3...\left(1\right) \\ \mathrm{At}x=3: \\ \frac{dy}{dx}=3{\left(3\right)}^2+2a\left(3\right)+b=0 \\ \Rightarrow 27+6a+b=0 \\ \Rightarrow -6a-b=27...\left(2\right) \\ \text{From eqs.}\left(\text{1}\right)\text{and}\left(\text{2}\right)\text{, we get} \\ a=-3,b=-9\mathrm{and}c\in R \end{gathered}$$