Question

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

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

Answer 1

Answer: (C)

$f\left(x\right)=\frac{4}{3}{x}^3-8x^2+16x+5\dots \left(1\right)$
Differentiating with respect to $x$, we get
$f^{\prime }\left(x\right)=\frac{4}{3}\times 3x^3-3x^3-16x+16$
Now for maximum/minimum we put $f^{\prime }\left(x\right)=0$
$\Rightarrow x^2-4x+4=0$
$\Rightarrow {\left(x-2\right)}^2=0$
$\Rightarrow x=2$
$f^{\prime \prime }\left(x\right)=8x-16$,
$f^{\prime \prime }\left(x\right){|}_{x=2}=0$
$f^{\prime \prime \prime }(x)=8\ne 0$
$\therefore x=2$ is the point of inflexion