Question

Consider the polynomial $p(x)=9x^8-18x^5-20x^3+15$. Then $p(x)=0$ has

• A. Exactly one real root in $\left[0,3^{1/3}\right]$
• B. Exactly two real roots in $\left[0,5^{1/5}\right]$
• C. At least two real roots in $\left[0,3^{1/3}\right]$
• D. No real roots in $\left[3^{1/5},5^{1/5}\right]$

Answer 1

Answer: (C)

Let $g(x)=\int p(x)dx$
$=x\left(x^8-3x^5-5x^3+15\right)+c$
$=x\left(x^3-3\right)\left(x^5-5\right)+c$
Clearly $f(0)=f\left(3^{1/3}\right)=f\left(5^{1/5}\right)=c$
$\therefore$ Rolle's theorem is applicable on $g(x)$ in intervals $\left[0,3^{1/3}\right],\left[3^{1/3},5^{1/5}\right]$ and in $\left[0,5^{1/5}\right]$
$\because y=x^{1/x}$ has a local maxima at $x=e$
$\therefore 0<5^{1/5}<3^{1/3}$
$\therefore$ There exists at least one ' $c$ ', each in $\left[0,5^{1/5}\right]$ and in $\left[5^{1/5},3^{1/3}\right]$ for which $g^{\prime }(x)=0$ i.e. $p(x)=0$
$\therefore p(x)=0$ has at least two real roots in $\left[0,3^{1/3}\right]$