Question

The function $f\left(x\right)=\{\begin{array}{c}ax(x-1)+bwhenx<1\\ x-1when1\le x\le 3\\ p{x}^2+qx+2whenx>3\end{array}$ Find the values of the constants $a,b,p,q$ so that $\left(i\right)f\left(x\right)$ is continuous for all $x$ $\left(ii\right)f^{\prime }\left(1\right)$ does not exist $\left(iii\right)f^{\prime }\left(x\right)$ is continuous at $x=3$

• A. $a=1,b=0,p=1/3,q=-1$
• B. $a\ne 1,b=0,p=1/3,q=-1$
• C. $a\ne 1,b=0,p=1/3,q=1$
• D. $a=1,b=0,p=1/3,q=1$

Answer 1

Answer: (B)

$a\ne 1,b=0,p=1/3,q=-1$

Making f(x) to be continuous at $x=1$ and $3$, we have
$a\left(0\right)+b=0$ and $3-1=9p+3q+2$
$\Rightarrow b=0$ and $q+3p=0$
Now, $f^{\prime }\left(x\right)=\{\begin{array}{c}2ax-a,x<1\\ 1,1\le x\le 3\\ 2px+q,x>3\end{array}$
The given condition implies $2a\left(1\right)-a\ne 1$ or $a\ne 1$
The last condition says $1=6p+q=6p-3p=3p$ (...substituting the value of q in terms of p)
$\Rightarrow p=\frac{1}{3}$ and $q=-1$