find the points of local Maxima and local minima,if any of the function f(x)=3x^4-2x^3-6x^2+6x+1.Also find the local maximum and local minimum value.
Answers
Answer:
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