Question

Does the function g(x) = -2x2 + 10x + 2 have a minimum or a maximum? Identify its value.
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Answer 1

maximum; 14.5

Explanation:

g(x) = −2x2 + 10x + 2

Determine whether the graph has a maximum or minimum value.

Since the a-value is negative, the graph opens downward and has a maximum value.

Find the x-value of the vertex.

Find the y-value of the vertex, g(2.5).

g(2.5) = −2(2.5)2 + 10(2.5) + 2

= −2(6.25) + 25 + 2

= −12.5 + 25 + 2

= 14.5

Therefore, the maximum value is 14.5.

Answer 2

(5/2, 29/2), and this represents a maximum

Explanation:

g(x) = -2x^2 + 10x + 2 is a quadratic function with coefficients {-2, 10, 2}.

The formula for the axis of symmetry (which passes through the vertex) is

x = -b/ [2a]. Here, a = -2; b = 10. Therefore, the axis of symmetry is

x = -10/ [2(-2)] = -10/(-4) = 5/2

Evaluate the function g(x) = -2x^2 + 10x + 2 at x = 5/2 to determine the y-coordinate of the vertex:

g(5/2) = -2(5/2)^2 + 10(5/2) + 2, or

g(5/2) = -2(25/4) + 25 + 2, or

g(5/2) = -25/2 + 27, or

g(5/2) = 14.5

The vertex is thus (5/2, 29/2), and this represents a maximum, because the coefficient of the x^2 term is negative (the curve opens downward).