The maximum slope of the curve

Question:

The maximum slope of the curve $y=\frac{1}{2} x^{4}-5 x^{3}+18 x^{2}-19 x$ occurs at the point:

  1. (1) $(2,9)$

     

  2. (2) $(2,2)$

  3. (3) $\left(3, \frac{21}{2}\right)$

  4. (4) $(0,0)$


Correct Option: , 2

Solution:

$\frac{d y}{d x}=2 x^{3}-15 x^{2}+36 x-19$

Let $f(x)=2 x^{3}-15 x^{2}+36 x-19$

$f^{\prime}(x)=6 x^{2}-30 x+36=0$

$x^{2}-5 x+6=0$

$x=2,3$

$f "(x)=12 x-30$

$f "(x)<0$ for $x=2$

At $x=2$

$y=8-40+72-38$

$y=72-70=2$

$\Rightarrow(2,2)$

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