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# Find the points at which the function f given by f (x) = (x - 2)4 (x + 1)3 has local maxima

13) 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

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Given function is
$f(x) = (x-2)^4(x+1)^3$
$f^{'}(x) = 4(x-2)^3(x+1)^3 + 3(x+1)^2(x-2)^4\\ f^{'}(x)= 0\\ 4(x-2)^3(x+1)^3 + 3(x+1)^2(x-2)^4=0\\ (x-2)^3(x+1)^2(4(x+1) + 3(x-2))\\ x = 2 , x = -1 \ and \ x = \frac{2}{7}$
Now, for value x close to $\frac{2}{7}$ and to the left of  $\frac{2}{7}$ ,  $f^{'}(x) > 0$ ,and for value close to $\frac{2}{7}$ and to the right of $\frac{2}{7}$  $f^{'}(x) < 0$
Thus,  point  x = $\frac{2}{7}$ is the point of maxima
Now, for value x close to 2 and to the Right of  2 ,  $f^{'}(x) > 0$ ,and for value close to 2 and to the left of 2  $f^{'}(x) < 0$
Thus, point x = 2 is the point of minima
There is no change in the sign when the value of x is -1
Thus x = -1 is the point of inflexion

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