Find the absolute maximum and absolute minimum values of f(x) = x⁴ − 8x² + 3 on [−1, 3].
How to lay out the answer. Follow the frame, then write it in your own words.
1 · Write what is given and what is to be found. f(x) = x⁴ − 8x² + 3 on [−1, 3]. To find: the absolute maximum and minimum values.
2 · Write the formula, rule or theorem. f is a polynomial, so it is continuous on [−1, 3]. Its extreme values occur at critical points inside the interval or at the end points.
3 · Find the critical points. f′(x) = 4x³ − 16x = 4x(x − 2)(x + 2) = 0 gives x = 0, 2 or −2. Reject x = −2, as it lies outside [−1, 3].
4 · Find f at the critical points and end points.
f(−1) = 1 − 8 + 3 = −4
f(0) = 3
f(2) = 16 − 32 + 3 = −13
f(3) = 81 − 72 + 3 = 12
5 · Compare the values. Largest value 12, smallest value −13.
6 · Check and box the answer. Absolute maximum 12 at x = 3; absolute minimum −13 at x = 2.
Scoring tip: List the values in one line and pick the largest and smallest. Say why x = −2 is dropped.
