Question

For the function $f(x)=\frac{4}{3}{x}^3-8x^2+16x+5,$ $x=2isapointof$

• A. local maxima
• B. local minima
• C. point of inflexion
• D. None of these

Answer 1

Answer: (C)

$f\left(x\right)=\frac{4}{3}{x}^3-8x^2+16x+5\dots \left(i\right)$
There will be a point of inflexion at point $x$
if ${\left(\frac{d^{2}y}{d{x}^2}\right)}_{at\left(x=2\right)}=0$, but $\frac{d^{3}y}{d{x}^3}\ne 0$
From Eq. $\left(i\right)$, $f^{\prime }\left(x\right)=4x^2-16x+16$
$f^{\prime \prime }\left(x\right)=8x-16$
$f^{\prime \prime }\left(2\right)=16-16=0$
and $f^{\prime \prime }\left(x\right)=8\ne 0$
Hence, at $x=2,f\left(x\right)$ shown point of inflexion