Find the local maximum and minimum values of f(x) = x^3 -3x^2-9x where x lies in [-4, 4].
- print Print
- list Cite
Expert Answers
calendarEducator since 2010
write12,554 answers
starTop subjects are Math, Science, and Business
The minimum and maximum values of f(x) = x^3 - 3x^2 - 9x have to be determined. At the points where the values are minimum and maximum f'(x) = 0. If f'(x) = 0 at x = c and f''(c) is negative there is a maximum value at c, else if f''(c) is positive there is a minimum value at x = c.
f(x) = x^3 - 3x^2 - 9x
f'(x) = 3x^2 - 6x - 9
Solving 3x^2 - 6x - 9 = 0
=> 3x^2 - 6x - 9 = 0
=> x^2 - 2x - 3 = 0
=> x^2 - 3x + x - 3 = 0
=> x(x - 3) + 1(x - 3) = 0
=> (x + 1)(x - 3) = 0
=> x = -1 and x = 3
Both x = -1 and x= 3 lie in [-4, 4].
The maximum value is f(-1) = 5 and the minimum value is f(-3) = -27.
It is not -76 as can be seen from the graph below:
The correct minimum value of the function for x in [-4, 4] is -27 and the maximum value is 5.
Related Questions
- Find the maximum or minimum value of f(x) = -3x^2 + 9x
- 2 Educator Answers
- Find the maximum or minimum value of f(x) = 2x^2 + 3x - 5
- 2 Educator Answers
- `f(x) = x + 1/x, [0.2, 4]` Find the absolute maximum and minimum values of f on the given...
- 1 Educator Answer
- `f(x) = x e^(-x^2/8), [-1, 4]` Find the absolute maximum and minimum values of f on the...
- 2 Educator Answers
- `f(x) = 3x^4 - 4x^3 - 12x^2 + 1, [-2, 3]` Find the absolute maximum and minimum values of f...
- 2 Educator Answers
calendarEducator since 2011
write5,348 answers
starTop subjects are Math, Science, and Business
You need to calculate the derivative of the function to find the maximum and minimum values of the function.
f'(x) = `3x^2 - 6x - 9`
You need to find the zeroes of derivative:
f'(x) = 0 => `3x^2 - 6x - 9` = 0 =>`x^2 - 2x - 3 ` = 0
`x^2 - 2x - 2 -1 = 0` => (x^2 - 1) - 2(x+1) = 0
(x-1)(x+1) - 2(x+1) = 0
Factoring out (x+1) yields:
(x+1)(x - 3) = 0 => x + 1 = 0 => x = -1
x - 3 = 0 => x = 3
The values of derivative of the function are negative between -1 and 3, hence the function decreases between -1 and 3.
The function attends its maximum value at x = -1 => f(-1) = -1 - 3 + 9 = 5.
The function attends its minimum value at x = 3 => f(3) = 27 - 27 - 27 = -27.