Trig/math problem?

Question

What number exceeds one-half its square by the greatest amount? I can't figure this out because of the way its worded. Somebody help please?

Answer 1

OK, let x be the number. Then "one half its square" is 1/2 x^2. So to calculate by how much x exceeds 1/2x^2, you just subtract:

d(x) = x - 1/2x^2

So you're looking for the value of x that maximizes d(x). This equation is quadratic, so it's a parabola; you just have to figure out what the vertex is.

d(x) = -1/2x^2 + x
= -1/2(x^2 - 2x)
= -1/2(x^2 - 2x +1) + 1/2
= -1/2(x - 1) + 1/2

So the vertex is the point (1, 1/2), and the value of x that exceeds one half its square by the greatest amount is 1.

Answer 2

1

Answer 3

1 because 1square is 1, then you half it, makes 0.5
so it exceeds half its square by 0.5

if you tried 10 though:
10 square is 100
half of 100 is 50
so ten does not exceed half its square

or 2:
2 square is 4
half of 4 is 2
so 2 is exactly the same as half of its square.

i hope this has helped you to understand the wording.

Answer 4

Turn this into an equation and graph it.

y = x - (1/2)x^2.

To find max and min take the derivative of the function.

dy/dx = 1 - x = 0
x = 1

The second derivative tells us if it it max or min.

d^2y/dx^2 = -1 < 0 so it's max

The answer is 1.