Mathematics - Functions Question with Solution | TestHub

$\begin{array}{rl}& f(x)=a{x}^3+b{x}^2+cx+41\\ & f^{\prime }(x)=3a{x}^2+2bx+cx\\ & \Rightarrow f^{\prime }(1)=3a+2b+c=2\dots (1)\end{array}$ $\begin{array}{rl}& {\mathrm{f}}^{\prime \prime }(\mathrm{n})=6\mathrm{ax}+2\mathrm{b}\\ & \Rightarrow {\mathrm{f}}^{\prime \prime }(1)=6\mathrm{a}+2\mathrm{b}=4\\ & 3\mathrm{a}+\mathrm{b}=2\dots \dots \dots .(2)\end{array}$ $\begin{array}{rl}& (1)-(2)\\ & b+c=\mathrm{0...}(3)\end{array}$ $\begin{array}{rl}& \mathrm{f}(1)=40\\ & \mathrm{a}+\mathrm{b}+\mathrm{c}+41=40\end{array}$ use (3) $a+41=40$ by (2) $\begin{array}{rl}& -3+b=2\Rightarrow b=5\&c=-5\\ & a^2+b^2+c^2=1+25+25=51\end{array}$