Find the points of local maxima and minima if any of the following function defined in $0 \leq x \leq 6, x^{3} - 6 x^{2} + 9 x + 15$

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Solution

Let $f (x) = x^{3} - 6 x^{2} + 9 x + 15$
$f^{'} (x) = 3 x^{2} - 12 x + 9 = 0 \Rightarrow x = 1, 3$
are the turning points
$f^{''} (x) = 6 x - 12$
$f^{''} (1) = - 6 < 0$
$\Rightarrow f (x)$ has maximum at $x = 1$
$f^{''} (3) = 6 > 0$
$\Rightarrow f (x)$ has maximumat $x = 3$
So, $x = 1$ is a point of local maxima & $x = 3$ is a point of local minima

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Find the points of local maxima and minima if any of the following function defined in 0≤x≤6,x3−6x2+9x+15

Let f(x)=x3−6x2+9x+15 f′(x)=3x2−12x+9=0⇒x=1,3 are the turning points f′′(x)=6x−12 f′′(1)=−6<0 ⇒f(x) has maximum at x=1 f′′(3)=6>0 ⇒f(x) has maximumat x=3 So, x=1 is a point of local maxima & x=3 is a point of local minima

Find the points of local maxima and minima if any of the following function defined in $0 \leq x \leq 6, x^{3} - 6 x^{2} + 9 x + 15$