I'm having problems solving this and showing the steps:
Find the absolute maximum and absolute minimum, if either exists, for
y = f(x)= x^3 - 12x + 12 -3 less equal to x greater than equal to 5
I'll rewrite the equation.
y = f(x) = x^3 -12x + 12
-3 <= x >= 5
dy/dx = 3x^2 -12
function is horizontal when dy/dx = 0
that is when
x^2 = 4
x = +/- 2
when x = -2
f(-2) = -8 + 24 + 12 = 28
and
d^2y/dx^2 = 6 x = -12 so a maximum
when x = 2
f(2) = -4
and
d^2y/dx^2 = 6x = +12 so a minimum
now check the end points
when x = -3
f(-3) = -27+36+12 = 21
when x=+5
f(5) = 125 - 60 +12 = 77
so the maximum is 77 at x = 5
and the minimum is -4 at x = 2
Thank you so much. This is a great help. I can see where I started to go wrong.
To find the absolute maximum and absolute minimum of a function on a given interval, you need to follow these steps:
1. Find the critical points inside the interval by taking the derivative of the function and setting it equal to zero.
2. Evaluate the function at the critical points and at the endpoints of the interval.
3. Compare the values obtained in step 2 to determine the absolute maximum and minimum.
Let's apply these steps to find the absolute maximum and absolute minimum for the given function, y = f(x) = x^3 - 12x + 12, on the interval -3 ≤ x ≤ 5.
Step 1: Find the critical points inside the interval.
To find the critical points, we need to take the derivative of the function and set it equal to zero:
f'(x) = 3x^2 - 12 = 0
Solving this equation gives us x = ±2.
Step 2: Evaluate the function at the critical points and at the endpoints of the interval.
Now, we need to evaluate the function at the critical points and the endpoints of the interval:
f(-3) = (-3)^3 - 12(-3) + 12 = 42
f(5) = (5)^3 - 12(5) + 12 = -13
f(2) = (2)^3 - 12(2) + 12 = -4
f(-2) = (-2)^3 - 12(-2) + 12 = 44
Step 3: Compare the values obtained in step 2.
Comparing the values, we can determine the absolute maximum and absolute minimum:
Absolute Maximum: The maximum value is 44, which occurs at x = -2.
Absolute Minimum: The minimum value is -13, which occurs at x = 5.
Therefore, the absolute maximum is 44 and occurs at x = -2, while the absolute minimum is -13 and occurs at x = 5.