Question

Отыскать наибольшее и наименьшее значение а) y=6x²+12x + 5,[-2;3] b)y=4x^5 - 5x^4 - 15,[1;1]

Answer 1

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Answer 2

To find the maximum and minimum values of the given functions within the specified intervals, we need to perform the following steps:

a) For the function y = 6x² + 12x + 5, on the interval [-2, 3]:

Step 1: Find the critical points by taking the derivative and setting it equal to zero. Step 2: Evaluate the function at the critical points and endpoints. Step 3: The largest value will be the maximum, and the smallest value will be the minimum.

Step 1: Find the derivative of y with respect to x: y' = d/dx(6x² + 12x + 5) y' = 12x + 12

Step 2: Find the critical points by setting y' equal to zero and solving for x: 12x + 12 = 0 12x = -12 x = -1

Step 3: Evaluate the function at the critical point and endpoints: y(-2) = 6(-2)² + 12(-2) + 5 = 24 - 24 + 5 = 5 y(-1) = 6(-1)² + 12(-1) + 5 = 6 - 12 + 5 = -1 y(3) = 6(3)² + 12(3) + 5 = 54 + 36 + 5 = 95

So, the maximum value is 95, and the minimum value is -1.

b) For the function y = 4x^5 - 5x^4 - 15, on the interval [1, 1]:

Since the interval [1, 1] consists of a single point (x = 1), there is no variation within the interval. Hence, the function's value at x = 1 will be both the maximum and minimum.

y(1) = 4(1)⁵ - 5(1)⁴ - 15 = 4 - 5 - 15 = -16

Therefore, for part b, the maximum and minimum values are both -16, which occurs at x = 1.