Recall that given the equation of the second degree (or quadratic)
ax ^ 2 + bx + c
Its solutions are:
x = (- b +/- root (b ^ 2-4ac)) / 2a
discriminating:
d = root (b ^ 2-4ac)
If d> 0, then the two roots are real (the radicand of the formula is positive).
If d = 0, then the root of the formula is 0 and, therefore, there is only one solution that is real and of multiplicity 2 (it is a double root).
If d <0, then the two roots are complex and, in addition, one is the conjugate of the other. That is, if one solution is x1 = a + bi, then the other solution is x2 = a-bi (we are assuming that a, b, c are real).
One solution:
A cut point with the x axis
Two solutions:
Two cutting points with the x axis.
Complex solutions:
Does not cut to the x axis
1) First derivative
dy / dx = 15x^4 - 75x^2 + 60
relative maxima => dy / dx = 0
=> 15x^4 - 75x^2 + 60 = 0
=> x^4 - 5x^2 + 4 = 0
Solve by factoring: (x^2 - 4) (x^2 - 1) = 0
=> x^2 = 4 => x = 2 and x = -2
x^2 = 1 => x = 1 and x = -1
So, there are four candidates -4, -1, 1 and 4.
To conclude whether they are relative maxima you must take the second derivative. Only those whose second derivative is negative are relative maxima.
2) Second derivative:
d^2 y / dx^2 = 60x^3 - 150x
Factor: 30x(2x^2 - 5)
The second derivative test tells that the second derivative in a relative maxima is less than zero.
=> 30x (2x^2 - 5) < 0
=> x < 0 and x^2 > 5/2
=> x < 0 and x < √(5/2) or x > √(5/2)
=> x = - 2 is a local maxima
Also, x > 0 and x^2 < 5/2 is a solution
=> x = 1 is other local maxima.
Those are the only two local maxima.
The answer is the option c) 2 local maxima.
Answer:
1) D: all real/(-inf, +inf) R: all real/(-inf, +inf)
2) x 5 or (-inf, 5)U(5,+inf)
Step-by-step explanation:
1) This is a linear equation, meaning that it has all real numbers all throughout.
This implies that you can plug in any x value and it will have a definite y value.
Domain is all reals or (-inf, +inf)
Range is all reals or (-inf, +inf)
2) We can see that if you plug in the number 5 for x, you would get 5-5 = 0 for the denominator. We can not divide a number by 0, therefore the domain is
x can not equal to 5 or (-inf, 5)U(5,+inf)
The answer is the first one , A
Answer:
The sum of x and y is a rational number
Step-by-step explanation:
X+Y or 5+7=12
A rational number is any number that can be divided without going on and on forever.