Question

Which of the following statements is/are true for the function $f\left(x\right)=x^{13}+7x^3-5x+1$

• A. $f\left(x\right)$ has at least $1$ root (real) in $[0,1]$
• B. $f^{\prime }\left(x\right)$ has at least $1$ root (real) in $R$
• C. $f\left(x\right)$ has at most $1$ root (real) in $[0,1]$
• D. $f^{\prime }\left(x\right)$ has at most $1$ real root in $R$

Answer 1

Answer: (A), (B)

The correct options are
A $f\left(x\right)$ has at least $1$ root (real) in $[0,1]$
B $f^{\prime }\left(x\right)$ has at least $1$ root (real) in $R$

$f\left(x\right)=x^{13}+7x^3-5x+1$
$f\left(0\right)=1(>0)$, $f\left(1\right)=1+7-5+1=4(>0)$
$f\left(\frac{1}{2}\right)={\left(\frac{1}{2}\right)}^{13}+\frac{7}{8}-\frac{5}{2}+1(<0)$
By mean value theorem $f\left(x\right)$ has atleast two roots in $(0,1)$.
$f^{{}^{\prime }}\left(x\right)=13x^{12}+21x^2-5$
$f^{{}^{\prime }}\left(0\right)=-5(<0)$, $f^{{}^{\prime }}\left(1\right)=13+21-5(>0)$
$f^{{}^{\prime }}(-1)=13+21-5(>0)$
By M.V.T. $f^{{}^{\prime }}\left(x\right)$ has atleast two roots in $R$.