Math Problem Statement

For g(x)=-3x^2+5.2x+7 find the maximum value of the fuction

Solution

To find the maximum value of the quadratic function g(x)=3x2+5.2x+7g(x) = -3x^2 + 5.2x + 7, we can use the vertex formula. The vertex of a parabola given by a quadratic function ax2+bx+cax^2 + bx + c occurs at x=b2ax = \frac{-b}{2a}.

Here, the coefficients are:

  • a=3a = -3
  • b=5.2b = 5.2
  • c=7c = 7

The x-coordinate of the vertex is:

x=b2a=5.22(3)=5.26=5.260.867x = \frac{-b}{2a} = \frac{-5.2}{2(-3)} = \frac{-5.2}{-6} = \frac{5.2}{6} \approx 0.867

Next, we substitute this xx-value back into the original function to find the maximum value:

g(0.867)=3(0.867)2+5.2(0.867)+7g(0.867) = -3(0.867)^2 + 5.2(0.867) + 7

Let's calculate it step by step:

  1. Calculate (0.867)2(0.867)^2:

(0.867)20.751(0.867)^2 \approx 0.751

  1. Substitute into the function:

g(0.867)=3(0.751)+5.2(0.867)+7g(0.867) = -3(0.751) + 5.2(0.867) + 7 g(0.867)=2.253+4.5084+7g(0.867) = -2.253 + 4.5084 + 7 g(0.867)9.2554g(0.867) \approx 9.2554

Thus, the maximum value of the function g(x)=3x2+5.2x+7g(x) = -3x^2 + 5.2x + 7 is approximately 9.255.

Would you like further details or have any other questions?


Here are 5 related questions:

  1. How does the sign of the coefficient aa in a quadratic function affect the shape of the parabola?
  2. What is the significance of the vertex in the context of maximizing or minimizing a quadratic function?
  3. How would the maximum value change if the coefficient of x2x^2 was positive?
  4. Can you find the roots of the function g(x)=3x2+5.2x+7g(x) = -3x^2 + 5.2x + 7?
  5. How would the function behave as xx approaches infinity?

Tip: In a quadratic function, if the coefficient of x2x^2 is negative, the parabola opens downward, and the vertex represents the maximum point.

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Math Problem Analysis

Mathematical Concepts

Quadratic Functions
Vertex Formula

Formulas

Vertex formula for quadratic functions

Theorems

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Suitable Grade Level

Grades 10-12