# 3. Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be: (v) $f (x) = x^3 - 6x^2 + 9x + 15$

Givrn function is
$f (x) = x^3 - 6x^2 + 9x + 15\\ f^{'}(x) = 3x^2 - 12x + 9\\ f^{'}(x)= 0\\ 3x^2 - 12x + 9 = 0\\ 3(x^2-4x+3)=0\\ x^2-4x+3 = 0\\ x^2 - x -3x + 3=0\\ x(x-1)-3(x-1) = 0\\ (x-1)(x-3) = 0\\ x=1 \ \ \ \ \ \ and \ \ \ \ \ \ \ x = 3$
Hence 1 and 3 are critical points
Now, we use the second derivative test
$f^{''}(x) = 6x - 12\\ f^{''}(1) = 6 - 12 = -6 < 0$
Hence, x = 1 is a point of maxima and the maximum value is
$f (1) = (1)^3 - 6(1)^2 + 9(1) + 15 = 1-6+9+15 = 19$
$f^{''}(x) = 6x - 12\\ f^{''}(3) = 18 - 12 = 6 > 0$
Hence, x = 1 is a point of minima and the minimum value is
$f (3) = (3)^3 - 6(3)^2 + 9(3) + 15 = 27-54+27+15 = 15$

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