Find the largest and smallest value of the function y = 4-3x + 2x ^ 2 on the segment [0; 3]
July 26, 2021 | education
| 1. Let’s find the first derivative of the function:
y ‘= (4 – 3x + 2x ^ 2)’ = -3 + 4x.
2. Let us equate this derivative to zero and find the critical points:
-3 + 4x = 0;
4x = 3;
x = 3: 4;
x = 3/4.
3. Find the value of the function at this point and at the ends of the given segment [0; 3]:
y (0) = 4 – 3 * 0 + 2 * 0 = 4;
y (3/4) = 4 – 3 * 3/4 + 2 * 9/16 = 4 – 9/4 + 9/8 = 4 – 18/8 + 9/8 = 4 – 9/8 = 4 – 1 1/8 = 4 – 1 – 1/8 = 3 – 1/8 = 2 + 1 – 1/8 = 2 + 8/8 – 1/8 = 2 7/8.
y (3) = 4 – 3 * 3 + 2 * 9 = 4 – 9 + 18 = 4 + 9 = 13.
Answer: fmax = 13, fmin = 2 7/8.
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