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g(x)=(5-2x)(14+2x) The function g is defined by the given equation. For what value of x does g(x) reach its maximum?

g(x)=(5−2x)(14+2x) g(x)=(5-2 x)(14+2 x) \newlineThe function g g is defined by the given equation. For what value of x x does g(x) g(x) reach its maximum?

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Q. g(x)=(5−2x)(14+2x) g(x)=(5-2 x)(14+2 x) \newlineThe function g g is defined by the given equation. For what value of x x does g(x) g(x) reach its maximum?
  1. Identify Function Type: Identify the type of function.\newlineThe function g(x)=(5−2x)(14+2x)g(x) = (5-2x)(14+2x) is a quadratic function because it can be written in the form ax2+bx+cax^2 + bx + c after expanding the product.
  2. Expand to Find Coefficients: Expand the function to find the coefficients.\newlineExpanding the product, we get:\newlineg(x)=(5)(14)+(5)(2x)−(2x)(14)−(2x)(2x)g(x) = (5)(14) + (5)(2x) - (2x)(14) - (2x)(2x)\newlineg(x)=70+10x−28x−4x2g(x) = 70 + 10x - 28x - 4x^2\newlineg(x)=−4x2−18x+70g(x) = -4x^2 - 18x + 70
  3. Write in Standard Form: Write the function in standard quadratic form.\newlineThe standard form of a quadratic function is ax2+bx+cax^2 + bx + c. From Step 22, we have:\newlineg(x)=−4x2−18x+70g(x) = -4x^2 - 18x + 70\newlineHere, a=−4a = -4, b=−18b = -18, and c=70c = 70.
  4. Determine Maximum X-Value: Determine the x-value at which the maximum occurs.\newlineFor a quadratic function in standard form ax2+bx+cax^2 + bx + c, the x-value of the vertex, which gives the maximum (if a < 0) or minimum (if a > 0) value, is given by −b/(2a)-b/(2a).\newlineHere, a=−4a = -4 and b=−18b = -18, so:\newlinex=−(−18)/(2×−4)x = -(-18) / (2 \times -4)\newlinex=18/−8x = 18 / -8\newlinex=−9/4x = -9/4
  5. Verify Maximum: Verify that this xx-value gives a maximum.\newlineSince the coefficient a=−4a = -4 is negative, the parabola opens downwards, and the vertex represents the maximum point of the function.

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