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Absolute extrema on a closed interval practice problems

The closed-interval method: check the critical numbers and the endpoints. 30 problems with worked solutions; in 30 of them every equation is proved by a computer algebra system.

Find the absolute maximum and minimum values of \( \displaystyle f(x) = x^{3} - 3 x + 4 \) on \( \displaystyle [-1, 1] \).

Problem 3.143medium✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x) = - x^{3} - \frac{3 x^{2}}{2} + 18 x + 3 \) on \( \displaystyle [-3, 2] \).

Problem 3.146medium✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x) = x^{3} + \frac{9 x^{2}}{2} + 6 x - 1 \) on \( \displaystyle [-4, 1] \).

Problem 3.46hard✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x) = x^{3} - 3 x \) on \( \displaystyle [-4, 3] \).

Problem 3.47hard✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x) = - x^{3} - \frac{3 x^{2}}{2} - 2 \) on \( \displaystyle [-4, 3] \).

Problem 3.48hard✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x) = - x^{3} + \frac{15 x^{2}}{2} - 18 x - 1 \) on \( \displaystyle [-4, 5] \).

Problem 3.49hard✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x) = - x^{3} + 3 x^{2} + 9 x - 4 \) on \( \displaystyle [-4, 4] \).

Problem 3.50hard✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x) = x^{3} - 12 x + 2 \) on \( \displaystyle [-4, 3] \).

Problem 3.51hard✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x) = - x^{3} - \frac{9 x^{2}}{2} - 6 x - 1 \) on \( \displaystyle [-3, 3] \).

Problem 3.52hard✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x) = - x^{3} + \frac{9 x^{2}}{2} + 1 \) on \( \displaystyle [-2, 4] \).

Problem 3.53hard✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x) = - x^{3} + 12 x - 4 \) on \( \displaystyle [-4, 5] \).

Problem 3.54hard✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x) = - x^{3} + 3 x + 3 \) on \( \displaystyle [-1, 2] \).

Problem 3.55hard✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x) = - x^{3} - \frac{9 x^{2}}{2} - 6 x \) on \( \displaystyle [-2, 3] \).

Problem 3.56hard✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x) = - x^{3} + 3 x^{2} + 9 x - 3 \) on \( \displaystyle [-2, 4] \).

Problem 3.57hard✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x) = x^{3} - \frac{15 x^{2}}{2} + 18 x - 2 \) on \( \displaystyle [-1, 3] \).

Problem 3.58hard✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x) = - x^{3} + \frac{9 x^{2}}{2} - 6 x + 3 \) on \( \displaystyle [-4, 5] \).

Problem 3.59hard✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x) = x^{3} + \frac{3 x^{2}}{2} - 18 x - 3 \) on \( \displaystyle [-4, 4] \).

Problem 3.60hard✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x) = x^{3} - 3 x^{2} + 3 \) on \( \displaystyle [-1, 2] \).

Problem 3.136hard✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x) = - x^{3} - 6 x^{2} - 9 x - 1 \) on \( \displaystyle [-4, 2] \).

Problem 3.137hard✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x) = - x^{3} + \frac{3 x^{2}}{2} + 6 x + 3 \) on \( \displaystyle [-3, 5] \).

Problem 3.138hard✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x) = x^{3} - \frac{3 x^{2}}{2} - 18 x - 3 \) on \( \displaystyle [-2, 4] \).

Problem 3.139hard✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x) = - x^{3} - 3 x^{2} - 2 \) on \( \displaystyle [-2, 3] \).

Problem 3.140hard✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x) = - x^{3} + 12 x + 1 \) on \( \displaystyle [-2, 4] \).

Problem 3.141hard✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x) = x^{3} - \frac{15 x^{2}}{2} + 18 x - 2 \) on \( \displaystyle [-2, 3] \).

Problem 3.142hard✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x) = - x^{3} - 3 x^{2} - 1 \) on \( \displaystyle [-2, 1] \).

Problem 3.144hard✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x) = - x^{3} - 3 x^{2} + 9 x - 2 \) on \( \displaystyle [-4, 4] \).

Problem 3.145hard✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x) = - x^{3} + 12 x - 3 \) on \( \displaystyle [-4, 3] \).

Problem 3.147hard✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x) = - x^{3} - \frac{3 x^{2}}{2} - 4 \) on \( \displaystyle [-3, 4] \).

Problem 3.148hard✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x) = - x^{3} + \frac{3 x^{2}}{2} + 2 \) on \( \displaystyle [-1, 1] \).

Problem 3.149hard✓ Every equation proved

Find the absolute maximum and minimum values of \( \displaystyle f(x) = x^{3} - \frac{15 x^{2}}{2} + 18 x - 4 \) on \( \displaystyle [-4, 3] \).

Problem 3.150hard✓ Every equation proved