(1 point) Is the graph of y=sin(x4) increasing or decreasing when x=14?
(enter increasing, decreasing, or neither).
Is it concave up or concave down?
(enter up, down, or neither).
To determine if the graph of y = sin(x^4) is increasing or decreasing when x = 14, we need to find the derivative of the function and evaluate it at x = 14.
Step 1: Find the derivative of y = sin(x^4) using the chain rule.
y' = cos(x^4) * (4x^3)
Step 2: Evaluate the derivative at x = 14.
y'(14) = cos(14^4) * (4 * 14^3)
Now, we can determine if the graph is increasing or decreasing when x = 14.
If y'(14) is positive, the graph is increasing at x = 14.
If y'(14) is negative, the graph is decreasing at x = 14.
If y'(14) is zero, i.e., y'(14) = 0, the graph has a critical point at x = 14, but we cannot determine if it is increasing or decreasing just based on this information.
Similarly, to determine if the graph is concave up or concave down, we need to find the second derivative and evaluate it at x = 14. Let's calculate that as well.
Step 3: Find the second derivative of y = sin(x^4).
To find the second derivative, we differentiate the first derivative obtained in step 1.
y'' = -4x^2 * sin(x^4) + cos(x^4) * (12x^2)
Step 4: Evaluate the second derivative at x = 14.
y''(14) = -4(14^2) * sin(14^4) + cos(14^4) * (12(14^2))
Now, we can determine if the graph is concave up or concave down at x = 14.
If y''(14) is positive, the graph is concave up at x = 14.
If y''(14) is negative, the graph is concave down at x = 14.
If y''(14) is zero, i.e., y''(14) = 0, the graph has an inflection point at x = 14, but we cannot determine if it is concave up or concave down just based on this information.
Now, you can calculate y'(14) and y''(14) using the given formulas and determine if the graph is increasing or decreasing when x = 14, and if it is concave up or concave down.
To determine whether the graph of y = sin(x^4) is increasing or decreasing at x = 14, we need to find the derivative of the function and evaluate it at x = 14.
Step 1: Find the derivative of y = sin(x^4) using the chain rule.
Let's denote the derivative of y with respect to x as y'.
Using the chain rule:
dy/dx = d/dx [sin(x^4)] = cos(x^4) * d/dx (x^4)
= cos(x^4) * 4x^3
Step 2: Evaluate the derivative at x = 14.
Substitute x = 14 into the derivative expression we found in step 1:
y'(14) = cos(14^4) * 4(14^3)
To determine whether the graph is increasing or decreasing at x = 14, we check the sign of the derivative value.
If y'(14) > 0, the graph is increasing at x = 14.
If y'(14) < 0, the graph is decreasing at x = 14.
If y'(14) = 0, we cannot determine the increasing or decreasing behavior.
To determine the concavity of the graph, we need to find the second derivative.
Step 3: Find the second derivative of y = sin(x^4).
Let's denote the second derivative as y''.
Using the chain rule again:
y'' = d/dx [cos(x^4) * 4x^3]
= [(d/dx) cos(x^4)] * 4x^3 + cos(x^4) * [(d/dx) 4x^3]
= -sin(x^4) * 4x^3 * 4x^3 + cos(x^4) * 12x^2
= -16x^6sin(x^4) + 12x^2cos(x^4)
Step 4: Evaluate the second derivative at x = 14.
Substitute x = 14 into the second derivative expression we found in step 3:
y''(14) = -16(14^6)sin(14^4) + 12(14^2)cos(14^4)
To determine whether the graph is concave up or concave down at x = 14, we check the sign of the second derivative value.
If y''(14) > 0, the graph is concave up at x = 14.
If y''(14) < 0, the graph is concave down at x = 14.
If y''(14) = 0, we cannot determine the concavity.
Please substitute x = 14 into the respective derivative and second derivative expressions to calculate the values.