Question

It is given that the Rolle's Theorem holds for the function $f\left(x\right)=x^3+{\mathrm{px}}^2+\mathrm{qx},x\in \left(1,2\right]$ at the point x = 4/3. The value of p + q is .

• A. 2
• B. 3
• C. -3
• D. 2

Answer 1

The correct option is B 3

It is given that the Rolle's Theorem holds for the function f(x) defined by $f\left(x\right)=x^3+{\mathrm{px}}^2+\mathrm{qx},x\in \left(1,2\right]$ with c = 4/3.
$\therefore f\left(1\right)=f\left(2\right)\mathrm{and}f\text{'}\left(c\right)=0\Rightarrow f\text{'}\left(\frac{4}{3}\right)=0\Rightarrow 1+p+q=8+4p+2q\mathrm{and}3{\left(\frac{4}{3}\right)}^2+2p\left(\frac{4}{3}\right)+q=0\Rightarrow 3p+q=-7\mathrm{and}\frac{16}{3}+\frac{8p}{3}+q=0\Rightarrow 3p+q=-7\mathrm{and}16+8p+3q=0\Rightarrow 3p+q=-7\mathrm{and}8p+3q=-16\mathrm{Solving}\mathrm{the}\mathrm{above}\mathrm{equations},\mathrm{we}\mathrm{get}p=-5\mathrm{and}q=8p+q=3.$