𝑓(𝑥)=3−3𝑥2,−5≤𝑥≤2
The absolute maximum value is
and this occurs at 𝑥 equals
The absolute minimum value is
and this occurs at 𝑥 equals
To find the absolute maximum and minimum values of the function 𝑓(𝑥)=3−3𝑥^2 over the interval −5≤𝑥≤2, we can follow these steps:
1. Find all critical points within the interval: Critical points occur where the derivative of the function is either zero or undefined. Differentiating the given function 𝑓(𝑥)=3−3𝑥^2, we get 𝑓'(𝑥)=−6𝑥. Setting 𝑓'(𝑥)=0, we find that the critical point occurs at 𝑥=0.
2. Evaluate the function at the endpoints of the interval: Evaluate 𝑓(𝑥) at 𝑥=−5 and 𝑥=2.
By substituting 𝑥=−5 into the function, we get 𝑓(−5) = 3 − 3(−5)^2 = 3 − 3(25) = 3 − 75 = −72.
By substituting 𝑥=2 into the function, we get 𝑓(2) = 3 − 3(2)^2 = 3 − 3(4) = 3 − 12 = −9.
3. Compare the values obtained from step 1 and step 2: The absolute maximum value is the largest value obtained from step 1 and step 2. The absolute minimum value is the smallest value obtained from step 1 and step 2.
Comparing the values, we find:
- The absolute maximum value is −9 (occurs at 𝑥=2).
- The absolute minimum value is −72 (occurs at 𝑥=−5).
Therefore, the absolute maximum value of the function 𝑓(𝑥)=3−3𝑥^2 is −9, which occurs at 𝑥=2. The absolute minimum value is −72, which occurs at 𝑥=−5.