find the turning points for the curve y=2x^3-1/2x^2-x+5. Determine the nature of the turning points.
Appreciate your help
dy/dx = 6x^2 - x - 1
= 0 at the turning points
(3x + 1)(2x - 1) = 0
x = -1/3 or x = 1/2
sub those values into original equation to get the whole points
y'' = 12x-1
when x = -1/3, y'' is negative, so x = -1/3 will yield a maximum point
when x = 1/2 , y'' is positive, , so x = 1/2 will yield a minimum point.
Well, I must say, finding turning points can be quite a "hair-raising" experience! But fear not, I'm here to assist you!
To find the turning points of the curve, we first need to find the derivative of the function. So, let's put on our mathematical wigs and differentiate the function y = 2x^3 - (1/2)x^2 - x + 5 with respect to x:
dy/dx = 6x^2 - x - 1.
To find the turning points, we need to determine the values of x where dy/dx equals zero. Let's set dy/dx to zero and solve for x:
6x^2 - x - 1 = 0.
Using the quadratic formula, x = (-b ± √(b^2 - 4ac)) / (2a), where a = 6, b = -1, and c = -1:
x = (1 ± √(1 - 4(6)(-1))) / (2(6)).
After simplifying, we get two possible values for x: x ≈ 1.33 and x ≈ -0.17.
Now, let's determine the nature of these turning points. To do this, we can take the second derivative. So, let's put on our detective hats and find the second derivative:
d^2y/dx^2 = 12x - 1.
Plugging in the values we found for x:
d^2y/dx^2 at x = 1.33 ≈ 14.96,
d^2y/dx^2 at x = -0.17 ≈ -3.04.
Since the second derivative at x = 1.33 is positive and the second derivative at x = -0.17 is negative, we can conclude that the turning point at x = 1.33 is a local minimum, while the turning point at x = -0.17 is a local maximum.
So, to summarize, the turning points for the curve y = 2x^3 - (1/2)x^2 - x + 5 are:
- A local minimum at (1.33, y) and
- A local maximum at (-0.17, y).
Remember, math can be a "rollercoaster" ride, but with a little humor, we can make it enjoyable!
To find the turning points of the curve and determine their nature, we need to find the first derivative and second derivative of the function y = 2x^3 - 1/2x^2 - x + 5.
Step 1: Find the first derivative (dy/dx) of the function.
To find the turning points, we need to find where the first derivative is equal to zero.
dy/dx = d/dx(2x^3 - 1/2x^2 - x + 5)
= 6x^2 - x - 1/2
Setting dy/dx = 0:
6x^2 - x - 1/2 = 0
Step 2: Solve the equation to find the x-coordinates of the turning points.
We can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. Let's use the quadratic formula.
By applying the quadratic formula, we have:
x = (-(-1) ± √((-1)^2 - 4(6)(-1/2))) / (2(6))
x = (1 ± √(1 + 12)) / 12
x = (1 ± √13) / 12
The x-coordinates of the turning points are (1 + √13) / 12 and (1 - √13) / 12.
Step 3: Find the y-coordinates of the turning points.
Plug the x-coordinates into the original equation y = 2x^3 - 1/2x^2 - x + 5 to get the corresponding y-values.
For x = (1 + √13) / 12:
y = 2((1 + √13) / 12)^3 - 1/2((1 + √13) / 12)^2 - ((1 + √13) / 12) + 5
y ≈ 4.712
For x = (1 - √13) / 12:
y = 2((1 - √13) / 12)^3 - 1/2((1 - √13) / 12)^2 - ((1 - √13) / 12) + 5
y ≈ 4.715
Step 4: Determine the nature of the turning points.
To determine the nature of the turning points, we need to find the second derivative (d²y/dx²).
Taking the derivative of dy/dx, we get:
d²y/dx² = d/dx(6x^2 - x - 1/2)
= 12x - 1
Evaluate the second derivative at the x-coordinates of the turning points found in Step 2.
For x = (1 + √13) / 12:
d²y/dx² = 12((1 + √13) / 12) - 1
= √13 - 1
≈ 2.605
For x = (1 - √13) / 12:
d²y/dx² = 12((1 - √13) / 12) - 1
= -√13 - 1
≈ -3.605
Since the second derivative at (1 + √13) / 12 is positive (approximately 2.605), the turning point at that x-coordinate is a local minimum.
Similarly, since the second derivative at (1 - √13) / 12 is negative (approximately -3.605), the turning point at that x-coordinate is a local maximum.
So, the nature of the turning points are as follows:
- (1 + √13) / 12: Local minimum
- (1 - √13) / 12: Local maximum
To find the turning points of the curve y = 2x^3 - (1/2)x^2 - x + 5 and determine their nature, we need to find the values of x where the slope of the curve changes sign. This can be done by finding the derivative of the function and setting it equal to zero.
Step 1: Find the derivative of the function y = 2x^3 - (1/2)x^2 - x + 5.
dy/dx = 6x^2 - x - 1
Step 2: Set dy/dx = 0 and solve for x.
6x^2 - x - 1 = 0
To solve this equation, you can use the quadratic formula which states that for an equation of the form Ax^2 + Bx + C = 0, the solutions for x are given by:
x = (-B ± √(B^2 - 4AC)) / (2A)
In our case, A = 6, B = -1, and C = -1. Substituting these values into the quadratic formula, we get:
x = (-(-1) ± √((-1)^2 - 4(6)(-1))) / (2(6))
x = (1 ± √(1 + 24)) / 12
x = (1 ± √25) / 12
x = (1 ± 5) / 12
So, the solutions for x are:
x₁ = (1 + 5) / 12 = 6/12 = 1/2
x₂ = (1 - 5) / 12 = -4/12 = -1/3
Step 3: Substitute these values of x back into the original function to find the corresponding y-values.
For x = 1/2:
y = 2(1/2)^3 - (1/2)(1/2)^2 - (1/2) + 5
y = 1/4 - 1/8 - 1/2 + 5
y = 5/4 - 1/2 - 1/2 + 20/4
y = 5/4 - 2/2 + 20/4
y = 5/4 - 4/4 + 20/4
y = 21/4
For x = -1/3:
y = 2(-1/3)^3 - (1/2)(-1/3)^2 - (-1/3) + 5
y = -2/27 - 1/54 + 1/3 + 5
y = -4/54 - 1/54 + 18/54 + 270/54
y = -5/54 + 18/54 + 270/54
y = 283/54
Step 4: The turning points are the coordinates (x₁, y₁) and (x₂, y₂), which are (1/2, 21/4) and (-1/3, 283/54).
To determine the nature of the turning points, we need to analyze the second derivative of the function. If the second derivative is positive at a turning point, it represents a local minimum, and if the second derivative is negative, it represents a local maximum.
Step 5: Find the second derivative of the function.
d²y/dx² = d/dx(6x^2 - x - 1)
d²y/dx² = 12x - 1
Step 6: Substitute the x-values of the turning points into the second derivative.
For x = 1/2:
d²y/dx² = 12(1/2) - 1
d²y/dx² = 6 - 1
d²y/dx² = 5 (positive)
For x = -1/3:
d²y/dx² = 12(-1/3) - 1
d²y/dx² = -4 - 1
d²y/dx² = -5 (negative)
Step 7: Interpretation of the results.
At the turning point (1/2, 21/4), the second derivative is positive (5), indicating a local minimum. This means that the curve is increasing and reaches a minimum at that point.
At the turning point (-1/3, 283/54), the second derivative is negative (-5), indicating a local maximum. This means that the curve is decreasing and reaches a maximum at that point.
Therefore, the nature of the turning points for the curve y = 2x^3 - (1/2)x^2 - x + 5 are a local minimum at (1/2, 21/4) and a local maximum at (-1/3, 283/54).