Question

find the maximum and minimum values of function 

$y = 2{x}^{3} - 15{x}^{2} + 36x + 1$

Answer 1

Answer:

To find the maximum and the minimum value of a function, we use the application of differentiation.

Step 1: Differentiate the given function once and equate it with zero.

Function: 2x³ - 15x² + 36x + 1

After differentiating, we get.

⇒ 3·2 x² - 2. 15x + 36

⇒ 6x² - 30x + 36  

Equating it with 0 we get,

⇒ 6x² - 30x + 36 = 0

Dividing by 6 as it is a common term we get,

⇒ x² - 5x + 6 = 0   →  Equation 1

Solving the quadratic equation we get,

⇒ x² - 2x - 3x + 6 = 0

⇒ x ( x - 2 ) -3 ( x - 2 ) = 0

⇒ ( x - 2 ) ( x - 3 ) = 0

⇒ x = 2, 3  ⇔ Roots of the Equation

Step 2: Differentiate Equation 1.

⇒ 2x - 5 → Equation 2

Step 3: Substitute the roots of Equation 1.

If the value of the equation comes to be a negative value, then the root taken is the maximum value of function. If the value comes to be positive after substituting , then the root substituted is the minimum value.

Substituting 2 in Equation 2, we get

⇒ 2 ( 2 ) - 5

⇒ 4 - 5 = -1

Since value is negative, 2 is the maximum value of the function.

Maximum Value = 2 ( 2 )³ - 15 ( 2 )² + 36 ( 2 ) + 1

⇒ Maximum value = 2 ( 8 ) - 15 ( 4 ) + 72 + 1

⇒ Maximum value = 16 - 60 + 73 = 29

Hence Maximum value of function is 29 at x = 2

Substituting 3 in equation 2 we get,

⇒ 2 ( 3 ) - 5

⇒ 6 - 5 = 1

Since value is positive, 3 is the minimum value of the function.

Minimum Value = 2 ( 3 )³ - 15 ( 3 )² + 36 ( 3 ) + 1

⇒ Minimum Value = 2 ( 27 ) - 15 ( 9 ) + 108 + 1

⇒ Minimum Value = 54 - 135 + 109 = 28

Hence minimum value of function is 28 at x = 3.

Answer 2

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