Question

Find the points at which the function $f$ given by $f\left(x\right)=(x-2{)}^4(x+1{)}^3$ has
(i) local maxima
(ii) local minima
(iii) point of reflexion.

Answer 1

Consider the given expression,

$f^{{}^{\prime }}\left(x\right)={(x-2)}^4{(x+1)}^3$ …….(1)

Differentiate with respect to x we get

$f^{{}^{\prime }}\left(x\right)={(x-2)}^{4}3{(x+1)}^2+{(x+3)}^{3}4{(x-2)}^3$ …(2)

For maxima and minima

$f^{\prime }\left(x\right)=0$

${(x-2)}^{4}3{(x+1)}^2+{(x+3)}^{3}4{(x-2)}^3=0$

${(x-2)}^3{(x+1)}^2[3(x-2)+(x+3)4]=0$

${(x-2)}^3{(x+1)}^2[3x-6+4x+12]=0$

${(x-2)}^3{(x+1)}^2(7x+6)=0$

$x=2,x=-1,x=\frac{-6}{7}$

At point $x=2,x=-1$ $f^{{}^{\prime }}\left(x\right)=0$ hence these are point of inflection.

And $x=\frac{-6}{7}$ ,$f^{{}^{\prime \prime }}\left(x\right)$ is positive ,then function will be maximum.so this is the point of local maxima.