Q.a.Find the stationary points on the graph of y=12x+3x^2-2x^3 and sketch the curve.
b.How does your sketch show that the equation 12x+3x^2-2x^3 =0 has exactly three real roots.
c.Use your graph to show that the eequation 12x+3x^2-2x^3 = -5 also has exactly three real roots.
d.For what range of values of k does the equation 12x+3x^2-2x^3 = k have i)exactly three real roots ii)only one real root?
a) y=12x+3x^2-2x^3
y' = 12+6x-6x^2 = 6(2-x)(1+x)
b) y(-3) > 0 and y(-1) < 0 and y(0) = 0 and y(1) > 0 and y(4) < 0
c) the line y = -5 also crosses the graph at three places
d) i) If the two stationary points are a min at y=a and a max at y=b, then a < k < b
ii) k < a or k > b
Can you write the answer of only d)
a. To find the stationary points on the graph of y = 12x + 3x^2 - 2x^3, we need to find the points where the derivative of y is equal to zero. Let's differentiate y with respect to x:
dy/dx = 12 + 6x - 6x^2
By setting the derivative equal to zero, we can find the values of x:
12 + 6x - 6x^2 = 0
Simplifying the equation, we get:
6x^2 - 6x - 12 = 0
Dividing through by 6, we have:
x^2 - x - 2 = 0
Factoring the equation, we get:
(x - 2)(x + 1) = 0
So, x can be either 2 or -1. These are the x-coordinates of the stationary points.
To find the corresponding y-values, we substitute the values of x back into the original equation:
For x = 2:
y = 12(2) + 3(2^2) - 2(2^3) = 24 + 12 - 16 = 20
For x = -1:
y = 12(-1) + 3(-1^2) - 2(-1^3) = -12 - 3 + 2 = -13
Therefore, the stationary points are (2, 20) and (-1, -13).
To sketch the curve, plot these points on a graph and draw a smooth curve passing through them.
b. The sketch of the curve shows that there are three real roots for the equation 12x + 3x^2 - 2x^3 = 0. This is because the curve intersects the x-axis at three distinct points, indicating three solutions for the equation where y = 0.
c. To show that the equation 12x + 3x^2 - 2x^3 = -5 also has exactly three real roots, we compare it to the equation 12x + 3x^2 - 2x^3 = 0. By subtracting -5 from both sides of the equation, we get:
12x + 3x^2 - 2x^3 + 5 = 0
This equation has the same form as the previous equation, just translated vertically by 5 units. Since the shape of the curve remains the same, the equation 12x + 3x^2 - 2x^3 = -5 will also have exactly three real roots.
d. For the equation 12x + 3x^2 - 2x^3 = k:
i) To find the range of values of k that will result in exactly three real roots, we observe the graph of the equation. By analyzing the curve, we can see that there will be three real roots when the horizontal line y = k intersects the curve at three distinct points. Therefore, the range of values of k that results in exactly three real roots is the interval between the lowest and highest points at which the horizontal line intersects the curve.
ii) For the equation to have only one real root, the horizontal line y = k should intersect the curve at only one point. This occurs when the line is tangent to the curve. So, the range of values of k that results in only one real root is the interval between the x-values where the horizontal line y = k is tangent to the curve.
a. To find the stationary points on the graph of the equation y = 12x + 3x^2 - 2x^3, we need to find the values of x where the derivative of y is equal to zero.
First, let's find the derivative of y with respect to x:
dy/dx = 12 + 6x - 6x^2
Now, set the derivative equal to zero and solve for x:
12 + 6x - 6x^2 = 0
This is a quadratic equation in terms of x. Use factoring, completing the square, or quadratic formula to find the values of x when dy/dx is equal to zero.
Once you have the values of x, substitute them back into the original equation y = 12x + 3x^2 - 2x^3 to get the corresponding y-values.
b. To show that the equation 12x + 3x^2 - 2x^3 = 0 has exactly three real roots, we can examine the graph of the equation. Sketch the curve using the values of x and y obtained in part a.
By analyzing the shape of the curve, you can see that it crosses the x-axis at three distinct points. This implies that the equation has three real roots.
c. To show that the equation 12x + 3x^2 - 2x^3 = -5 also has exactly three real roots, we can make a slight adjustment. Rearrange the equation as follows:
12x + 3x^2 - 2x^3 + 5 = 0
Now, the equation is in the same form as before, but with a different constant term. Follow the same steps as in part b to sketch the curve and observe that it crosses the x-axis at three distinct points.
d. To find the range of values of k for which the equation 12x + 3x^2 - 2x^3 = k has exactly three real roots or only one real root, we can use the discriminant.
i) For exactly three real roots, the discriminant of the equation 12x + 3x^2 - 2x^3 = k must be greater than zero. By calculating the discriminant, which is based on the coefficients of x and x^2 terms, you can determine the range of k values that satisfy this condition.
ii) For only one real root, the discriminant of the equation must be equal to zero. Again, calculate the discriminant based on the coefficients of x and x^2 terms to determine the range of k values that satisfy this condition.
By finding the discriminant and evaluating the range of k values, you can determine the conditions under which the equation has either three real roots or only one real root.