Question

If $f\left(x\right)=(x-3{)}^5(x+1{)}^4,$ then which of the following is/are NOT CORRECT ?

• A. $x=\frac{7}{9}$ is a point of local maxima
• B. $x=3$ is a point of local minimum
• C. $f$ has no point of local maxima
• D. $x=-1$ is a point of local maxima

Answer 1

Answer: (A), (B), (C)

$f$ has no point of local maxima

$f\left(x\right)=(x-3{)}^5(x+1{)}^4$
$\Rightarrow f^{\prime }\left(x\right)=4(x-3{)}^5(x+1{)}^3+5(x+1{)}^4(x-3{)}^4$
$=(x-3{)}^4(x+1{)}^3(4x-12+5x+5)$
$=(x-3{)}^4(x+1{)}^3(9x-7)$
From $f^{\prime }\left(x\right)=0$
$x=-1,3,\frac{7}{9}$

From the above sign change of $f^{\prime }\left(x\right)$
$x=-1$ is the point of local maxima,
$x=\frac{7}{9}$ is the point of local minima,
At $x=3$ neither minima nor maxima.