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Tell whether the function has a minimum value or a maximum value. Then find the value.

19. f(x) = -4x2² - 24x + 15


20. f(x) = 6x² + 36x - 20

User Sam Woods
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Answer: To determine whether the given functions have a minimum value or a maximum value, we can use the concept of concavity and the coefficient of the leading term.

  • f(x) = -4x^2 - 24x + 15

The coefficient of the leading term (-4x^2) is negative, which means the parabola opens downwards. In this case, the function has a maximum value. To find the value, we need to locate the vertex of the parabola. The vertex of a parabola in the form f(x) = ax^2 + bx + c is given by the coordinates (h, k), where h = -b / 2a and k is the maximum (or minimum) value of the function.

For f(x) = -4x^2 - 24x + 15:

a = -4, b = -24

h = -(-24) / (2 * (-4)) = 24 / 8 = 3

Now, substitute h back into the function to find the maximum value (k):

f(3) = -4(3)^2 - 24(3) + 15

f(3) = -4(9) - 72 + 15

f(3) = -36 - 72 + 15

f(3) = -93

So, the function f(x) = -4x^2 - 24x + 15 has a maximum value of -93.

  • f(x) = 6x^2 + 36x - 20

The coefficient of the leading term (6x^2) is positive, which means the parabola opens upwards. In this case, the function has a minimum value. Again, we can find the value by locating the vertex of the parabola.

For f(x) = 6x^2 + 36x - 20:

a = 6, b = 36

h = -36 / (2 * 6) = -36 / 12 = -3

Now, substitute h back into the function to find the minimum value:

f(-3) = 6(-3)^2 + 36(-3) - 20

f(-3) = 6(9) - 108 - 20

f(-3) = 54 - 108 - 20

f(-3) = -74

So, the function f(x) = 6x^2 + 36x - 20 has a minimum value of -74.

User Gbc
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