Physics, asked by Vishal101100, 10 months ago

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Answered by CunningKing

• We know that,

x³/3 = 3x²/3

= x²

• Similarly,

5x²/2 = 2*5x/2

= 5x

• 6x is a constant. So, its differentiation becomes 0.

Step - I :-

Now,

y = x³/3 + 5x²/2 + 6x

⇒dy/dx = y' = x² + 5x + 0 = 0

⇒y' = x² + 5x = 0

⇒y' = x(x + 5) = 0

⇒ x = 0, or x = -5

Step - II :-

y" = dy'/dx = x² + 5x

⇒y" = 2x + 5

⇒y" = 2(0) + 5 or y" = 2(-5) + 5

⇒y" = 0 + 5 or y" = -10 + 5

⇒y" = 5 or y" = -5

5 > 0 and -5 < 0

So, 5 is minima and -5 is maxima.

$\sf{y_{max}=\frac{(2)^3}{5}+\frac{5(2)^2}{2}+6(2) }\\\\\sf{\implies y_{max}=\frac{8}{5}+\frac{20}{2}+12 }\\\\\sf{\implies y_{max}=\frac{236}{10} }\\\\\sf{\boxed{\implies y_{max}=23.6}}$

$\sf{y_{min}=\frac{(0)^3}{3}+\frac{5(0)^2}{2}+6(0) }\\\\\sf{\implies y_{min}=0+0+0}\\\\\sf{\boxed{\implies y_{min}=0}}$

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