Find the range of all possible values of B if the graph of
P(x) = 12x⁴ - 5x³ -38x² + Bx + 6
crosses the x-axis between 0 and -1
12x^4 - 5x^3 - 38x^2 + 0x + 6 = 0
P(0) = 6
P(-1) = -B-15
So, B > -15
Could you explain what you did in the first and the last line of your comment?
To find the range of all possible values of B, we need to determine when the graph of the polynomial function P(x) = 12x⁴ - 5x³ - 38x² + Bx + 6 crosses the x-axis between 0 and -1.
When a polynomial crosses the x-axis, it means that there is at least one real root (x-intercept) for that polynomial. In other words, there is a value of x for which P(x) = 0.
To determine when the graph crosses the x-axis between 0 and -1, we need to find the values of B for which the polynomial has a real root between 0 and -1.
We can use the concept of the Intermediate Value Theorem to determine this. According to the Intermediate Value Theorem, if a function is continuous on a closed interval [a, b], and f(a) and f(b) have opposite signs, then there exists at least one value c in the interval (a, b) such that f(c) = 0.
In our case, the interval is [0, -1]. We need to check if the polynomial P(x) changes sign between 0 and -1.
Let's substitute x = 0 into the polynomial:
P(0) = 12(0)⁴ - 5(0)³ - 38(0)² + B(0) + 6 = 6
Let's substitute x = -1 into the polynomial:
P(-1) = 12(-1)⁴ - 5(-1)³ - 38(-1)² + B(-1) + 6 = 12 - 5 + 38 + (-B) + 6 = 51 - B
Now, we need to determine the sign change between P(0) = 6 and P(-1) = 51 - B.
If P(0) = 6 and P(-1) = 51 - B have opposite signs, then there exists at least one value c in the interval (0, -1) where P(c) = 0.
If 6 and 51 - B have opposite signs, it means that 6 is positive and 51 - B is negative, or vice versa.
To check for the range of possible values of B, we need to find where the sign changes between the two expressions:
Case 1: If 6 is positive and 51 - B is negative:
6 > 0 and 51 - B < 0
Simplifying the inequality: 51 < B
Case 2: If 6 is negative and 51 - B is positive:
6 < 0 and 51 - B > 0
Simplifying the inequality: B < 51
The possible range of values for B is the intersection of both cases:
51 < B and B < 51
Therefore, the range of all possible values of B is B < 51.