Find the local maxima or local minima of $f (x) = x^{3} - x^{2} + 9 x + 15$ . Also find the local maximum or local minimum values as the case may be.

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Solution

$f (x) = x^{3} - x^{2} + 9 x + 15$
$\frac{d y}{d x} = 3 x^{2} - 2 x + 9 \frac{d y}{d x} = 0 3 x^{2} - 2 x + 9 = 0 D = 4 - 4 (3 \times 9) D < 0$
hence $\frac{d y}{d x} > 0$ for all values of $x$
$f (x)$ function is always an increasing funtion
hence $n o m a x i m a o r m i n i m a$

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$f (x) = x^{3} - x^{2} + 9 x + 15$
$\frac{d y}{d x} = 3 x^{2} - 2 x + 9 \frac{d y}{d x} = 0 3 x^{2} - 2 x + 9 = 0 D = 4 - 4 (3 \times 9) D < 0$
hence $\frac{d y}{d x} > 0$ for all values of $x$
$f (x)$ function is always an increasing funtion
hence $n o m a x i m a o r m i n i m a$