Question
y=2x^3-5x^2-3x
how many turns are there
how many turns are there
Answers
GPT-4o mini
To determine the number of turns or local extrema of the polynomial function \( y = 2x^3 - 5x^2 - 3x \), we need to find the critical points by taking the derivative and setting it equal to zero.
1. **Calculate the first derivative**:
\[
y' = \frac{d}{dx}(2x^3 - 5x^2 - 3x) = 6x^2 - 10x - 3
\]
2. **Set the first derivative to zero to find critical points**:
\[
6x^2 - 10x - 3 = 0
\]
3. **Use the quadratic formula to solve for \( x \)**:
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Where \( a = 6 \), \( b = -10 \), and \( c = -3 \).
Calculating the discriminant:
\[
b^2 - 4ac = (-10)^2 - 4 \cdot 6 \cdot (-3) = 100 + 72 = 172
\]
Since the discriminant is positive, there will be two distinct real solutions for \( x \).
4. **Finding the number of turns**:
Since there are two distinct critical points, the polynomial will have two local extrema (turns).
Thus, the function \( y = 2x^3 - 5x^2 - 3x \) has **two turns**.
1. **Calculate the first derivative**:
\[
y' = \frac{d}{dx}(2x^3 - 5x^2 - 3x) = 6x^2 - 10x - 3
\]
2. **Set the first derivative to zero to find critical points**:
\[
6x^2 - 10x - 3 = 0
\]
3. **Use the quadratic formula to solve for \( x \)**:
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Where \( a = 6 \), \( b = -10 \), and \( c = -3 \).
Calculating the discriminant:
\[
b^2 - 4ac = (-10)^2 - 4 \cdot 6 \cdot (-3) = 100 + 72 = 172
\]
Since the discriminant is positive, there will be two distinct real solutions for \( x \).
4. **Finding the number of turns**:
Since there are two distinct critical points, the polynomial will have two local extrema (turns).
Thus, the function \( y = 2x^3 - 5x^2 - 3x \) has **two turns**.