Rogelio is asked to sketch a graph of g(x)=3x3(x−5)2(5−x)4 . How many turning points should his sketch include?(1 point)
1. three at most
2. exactly eight
3. eight at most
4. nine at most
To determine the number of turning points, we need to find the number of times the graph changes from increasing to decreasing or vice versa. This occurs at the points where the derivative of the function is equal to zero.
Taking the derivative of g(x):
g'(x) = 3 * (x-5)^2 * (5-x)^4 * [3x^2 - 2x(5+x) + 2(x-5)(5-x)].
We can ignore the constant coefficient and focus on the quantity inside the square brackets, which is a quadratic equation:
3x^2 - 2x(5+x) + 2(x-5)(5-x)
Expanding this equation, we get:
3x^2 - 2x(5+x) + 2(25 - 10x - 5x + x^2)
= 3x^2 - 2x^2 - 10x - 2x^2 + 10x - 50 + 20x - 10x
= -x^2 + 20x - 50
Setting this equal to zero and factoring, we get:
(-x + 10)(x - 5) = 0
So, x = 10 or x = 5.
Now, using the second derivative test, we can determine the nature of these points.
Taking the second derivative of g(x):
g''(x) = 3 * (x-5)^2 * (5-x)^4 * [6x - 2(5+x) + 2(x-5)(-1)]
= 3 * (x-5)^2 * (5-x)^4 * [6x - 10 - 2x -10 + 2x -10]
= 3 * (x-5)^2 * (5-x)^4 * [6x - 30]
Plugging in x = 10:
g''(10) = 3 * (10-5)^2 * (5-10)^4 * [6*10 - 30]
= 3 * 5^2 * (-5)^4 * [60 - 30]
= 3 * 25 * 625 * 30
= 1406250 > 0
Plugging in x = 5:
g''(5) = 3 * (5-5)^2 * (5-5)^4 * [6*5 - 30]
= 0
Since the second derivative is positive at x = 10 and the second derivative is zero at x = 5, we have a local minimum and an inflection point at x = 10.
Therefore, the graph of g(x) has at least one turning point, and the answer is:
1. three at most
To determine the number of turning points for the graph of g(x) = 3x^3(x−5)^2(5−x)^4, we need to consider the behavior of the function as x approaches different values.
The exponent of 3 in g(x) means that the graph could potentially have three turning points: one near x = -∞, one near x = 0, and one near x = +∞.
The first factor, (x−5)^2, is squared and always positive, so it does not change the number of turning points.
The second factor, (5−x)^4, is raised to an even power, which means it is always positive and does not change the number of turning points.
Therefore, the function g(x) will have at most three turning points.
Hence, the correct answer is:
1. three at most
To determine the number of turning points on a graph, we need to consider the degree of the polynomial function.
In this case, the function g(x) has a degree of 3x^3 multiplied by (x-5)^2 multiplied by (5-x)^4.
The highest degree term in the function is 3x^3, meaning the graph will have three "arms" or "branches" that gradually curve or turn.
Since each of these branches represents a turning point, the graph of g(x) should have exactly three turning points.
Therefore, the correct answer is 1. Three at most.