Answer:

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Step-by-step explanation:

MAXIMUM or minimum can be seen by using derivatives.

Steps1: First FIND first derivative of the function

Step2: Put it equal to zero and find x were first derivative is zero

Step3: Now find SECOND derivative

Step4: Put x for which first derivative was zero in equation of second derivative

Step5: If second derivative is GREATER than zero then function takes minimum value at that x and if second derivative is negative then function will take maximum value at that x. If Second derivative is zero them it means that this is the point of inflection.

f

′

(x)=3x

2

−12x+9

Putting this equal to zero, we get

f

′

(x)=0

3x

2

−12x+9=0

⇒(x−1)(x−3)=0

⇒x=1,3

Now let's see the DOUBLE derivative of this function.

f

′′

(x)=6x−12

At x=1

f

′′

(1)=6×1−12=−6

So function will take maximum value at x=1, which is given by

f(1)=19

At x=3

f

′′

(3)=6×3−12=6

This is positive at x=3, so function will take a minimum value at x=3.

Minimum value is given by f(3)=3

3

−6×3

2

+9×3+15=15

Minimum value of the function is 15

Maximum value of the function is 19