Question

Find the points at which the function $f$ given by $f(x)=(x-2)^4(x+1)^3$ has (i)local maxima (ii)local minima (iii)point of inflexion

Answer 1

The given function is $f(x)=(x-2)^4(x+1)^3$
$∴f'(x)=4(x-2)^3(x+1)^3+3(x+1)^2(x-2)^4$
$=(x-2)^3(x+1)^2[4(x+1)+3(x-2)]$
$=(x-2)^3(x+1)^2(7x-2)$
Now,$f'(x)=0$
$⇒x=-1$ and $x=\frac{2}{7}\, or\, x=2$
Now, for values of $x$ close to $\frac{2}{7}$ and to the left of $\frac{2}{7}\, f'(x)>0.$ Also, for values of $x$ close to $\frac{2}{7}$ and to the left of $\frac{2}{7}\, f'(x)<0.$
Thus $x=\frac{2}{7}$ is the point of local maxima.
Now, for values of $x$ close to $2$ and to the left of $2,f'(x)<0.$
Also, for values of $x$ close to $2$ and to the right of $2, f'(x)>0$.
Thus, $x = 2$ is the point of local minima.
Now, as the value of $x$ varies through $−1,F'(X)$ does not changes its sign. Thus, $x=−1$ is the point of inflexion.