F(x) = 18-x^2 is a parabola having vertex at (0, 18) and opening downwards.
g(x) = 2x^2-9 is a parabola having vertex at (0, -9) and opening upwards.
By symmetry, let the x-coordinates of the vertices of rectangle be x and -x => its width is 2x.
Height of the rectangle is y1 + y2, where y1 is the y-coordinate of the vertex on the parabola f and y2 is that of g.
=> Area, A
= 2x (y1 - y2)
= 2x (18 - x^2 - 2x^2 + 9)
= 2x (27 - 3x^2)
= 54x - 6x^3
For area to be maximum, dA/dx = 0 and d²A/dx² < 0
=> 54 - 18x^2 = 0
=> x = √3 (note: x = - √3 gives the x-coordinate of vertex in second and third quadrants)
d²A/dx² = - 36x < 0 for x = √3
=> maximum area
= 54(√3) - 6(√3)^3
= 54√3 - 18√3
= 36√3.
3
is the simplify expression
Step-by-step explanation:
Be sure to include the "=" sign: <span>f(x) = 2^x -7
To find the x-intercepts, set f(x) = 0 and solve for x: 2^x - 7 = 0,
or
2^x = 7
Take the common log of both sides: x log 2 = log 7
log 7
Solve for x: x = --------- = 2.81 (approx)
log 2
</span>
Step-by-step explanation:
19 is the greatest value
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