Practicing Success
Rolles's theorem holds for the function $x^3+\mathrm{b} x^2+\mathrm{c} x, 1 \leq x \leq 2$ at the point $\frac{4}{3}$, the respective values of $b$ and $c$ are : |
$-5,8$ $5,-8$ $8,-5$ $-5,-8$ |
$-5,8$ |
$f(x)=x^3+b x^2+c x \quad 1 \leq x \leq 2$ So by Rolle's theorem f(x) is continous on [1, 2] differentiable on (1, 2) Hence $f(1)=f(2)$ $\Rightarrow (1)^3+b(1)^2+c(1)=(2)^3+(2)^2 b+(2) c$ $1+b+c=8+4 b+2 c$ $(8-1)+(4 b-b)+(2 c-c)=0$ 7 + 3b + c = 0 .....(1) $f'(x)=\frac{d}{d x}\left(x^3+b x^2+c\right)=3 x^2+2 b x+c-2$ at $4 / 3 \quad f'(4 / 3)=0$ as por rolle's theorem f'(4/3) = 0 substituting from eq (2) $=3 \times\left(\frac{4}{3}\right)^2+2 \times \frac{4}{3} b+c=0$ $\Rightarrow 3 \times \frac{16}{9}+\frac{8}{3} b+c=0$ $=\frac{16}{3}+\frac{8}{3} b+c=0$ $16+8 b+3 c=0$ ........(3) eq (1) × (2) --- eq (3) $3 \times(7+3 b+c=0)$ $-(16+8 b+3 c=0)$ $\Rightarrow 21+a b+3 c=0$ $-16+8 b+3 c=0 $ $5+b=0 \Rightarrow b =-5$ Sunstituting b = -5 in eq (1) → we get $7 \times 3(-5)+c=0$ $7-15+c=0 \Rightarrow-8+c=0 \Rightarrow c=8$ b = -5, c = 8 |