The local minimum value of the function f(x) = 6 + 9x² - 6x³ is 6 and the local maximum value of the function f(x) = 6 + 9x² - 6x³ is 9.
For given question,
We have been given a function f(x) = 6 + 9x² - 6x³
We need to find the local maximum and local minimum of the function f(x)
First we find the first derivative of the function.
⇒ f'(x) = 0 + 18x - 18x²
⇒ f'(x) = - 18x² + 18x
Putting the first derivative of the function equal to zero, we get
⇒ f'(x) = 0
⇒ - 18x² + 18x = 0
⇒ 18(-x² + x) = 0
⇒ x (-x + 1) = 0
⇒ x = 0 or -x + 1 = 0
⇒ x = 0 or x = 1
Now we find the second derivative of the function.
⇒ f"(x) = - 36x + 18
At x = 0 the value of second derivative of function f(x),
⇒ f"(0) = - 36(0) + 18
⇒ f"(0) = 0 + 18
⇒ f"(0) = 18
Here, at x=0, f"(x) > 0
This means, the function f(x) has the local minimum value at x = 0, which is given by
⇒ f(0) = 6 + 9(0)² - 6(0)³
⇒ f(0) = 6 + 0 - 0
⇒ f(0) = 6
At x = 1 the value of second derivative of function f(x),
⇒ f"(1) = - 36(1) + 18
⇒ f"(1) = - 18
Here, at x = 1, f"(x) < 0
This means, the function f(x) has the local maximum value at x = 1, which is given by
⇒ f(1) = 6 + 9(1)² - 6(1)³
⇒ f(1) = 6 + 9 - 6
⇒ f(1) = 9
So, the function f(x) = 6 + 9x² - 6x³ has local minimum at x = 0 and local maximum at x = 1.
Therefore, the local minimum value of the function f(x) = 6 + 9x² - 6x³ is 6 and the local maximum value of the function f(x) = 6 + 9x² - 6x³ is 9.
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