Let g be the function defined by g(x)=∫x−1(−12+cos(t3+2t))ⅆt for 0
To find the value of x at which the function g attains a relative maximum, we need to analyze the behavior of g(x) by taking its derivative and finding the critical points
To find the value of x at which the function g attains a relative maximum, we need to analyze the behavior of g(x) by taking its derivative and finding the critical points.
First, let’s find the derivative of g(x) using the Fundamental Theorem of Calculus and the Chain Rule:
g'(x) = d/dx [∫x^-1 (-12 + cos(t^3 + 2t)) dt]
Since we are differentiating with respect to x, we need to apply the Chain Rule to the integrand, treating t as a function of x. Let’s define another function u = t^3 + 2t, so we have:
g'(x) = d/dx [∫x^-1 (-12 + cos(u)) du/dx] (by Chain Rule)
Now, since the limits of integration involve x, we need to apply the Fundamental Theorem of Calculus again and evaluate the derivative inside the integral:
g'(x) = ∫x^-1 (du/dx) (-12 + cos(u)) du
Next, we can simplify this expression further. The derivative of u = t^3 + 2t with respect to x is:
du/dx = 3t^2 + 2
Substituting this back into the previous expression for g'(x):
g'(x) = ∫x^-1 (3t^2 + 2)(-12 + cos(u)) du
Now, to find the critical points, we set g'(x) equal to zero:
g'(x) = ∫x^-1 (3t^2 + 2)(-12 + cos(u)) du = 0
Since we are looking for a relative maximum, we also need to check the endpoints of the given interval, 0 < x < π/2. So let's consider x = 0 and x = π/2 separately as well.
For x = 0:
g(0) = ∫0^-1 (-12 + cos(u)) du = ∫1^∞ (-12 + cos(u)) du
For x = π/2:
g(π/2) = ∫(π/2)^-1 (-12 + cos(u)) du = ∫2/π (-12 + cos(u)) du
Now, we need to evaluate the definite integrals for the endpoints separately. Note that the integral from 1 to infinity is improper, but since the function is continuous on the given interval, we can determine its behavior.
After evaluating the definite integrals, we compare the values of g(0), g(π/2), and the critical points of g(x) to identify the x-value at which g attains a relative maximum. The x-value corresponding to the highest function value will be the answer.
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To find the value of x at which the function g attains a relative maximum, we need to analyze the behavior of g(x) by taking its derivative and finding the critical points
To find the value of x at which the function g attains a relative maximum, we need to analyze the behavior of g(x) by taking its derivative and finding the critical points.
First, let’s find the derivative of g(x) using the Fundamental Theorem of Calculus and the Chain Rule:
g'(x) = d/dx [∫x^-1 (-12 + cos(t^3 + 2t)) dt]
Since we are differentiating with respect to x, we need to apply the Chain Rule to the integrand, treating t as a function of x. Let’s define another function u = t^3 + 2t, so we have:
g'(x) = d/dx [∫x^-1 (-12 + cos(u)) du/dx] (by Chain Rule)
Now, since the limits of integration involve x, we need to apply the Fundamental Theorem of Calculus again and evaluate the derivative inside the integral:
g'(x) = ∫x^-1 (du/dx) (-12 + cos(u)) du
Next, we can simplify this expression further. The derivative of u = t^3 + 2t with respect to x is:
du/dx = 3t^2 + 2
Substituting this back into the previous expression for g'(x):
g'(x) = ∫x^-1 (3t^2 + 2)(-12 + cos(u)) du
Now, to find the critical points, we set g'(x) equal to zero:
g'(x) = ∫x^-1 (3t^2 + 2)(-12 + cos(u)) du = 0
Since we are looking for a relative maximum, we also need to check the endpoints of the given interval, 0 < x < π/2. So let's consider x = 0 and x = π/2 separately as well. For x = 0: g(0) = ∫0^-1 (-12 + cos(u)) du = ∫1^∞ (-12 + cos(u)) du For x = π/2: g(π/2) = ∫(π/2)^-1 (-12 + cos(u)) du = ∫2/π (-12 + cos(u)) du Now, we need to evaluate the definite integrals for the endpoints separately. Note that the integral from 1 to infinity is improper, but since the function is continuous on the given interval, we can determine its behavior. After evaluating the definite integrals, we compare the values of g(0), g(π/2), and the critical points of g(x) to identify the x-value at which g attains a relative maximum. The x-value corresponding to the highest function value will be the answer.
More Answers:
Analyzing Functions | Finding and Understanding Critical Numbers in CalculusAnalyzing Local Maxima | Understanding the Highest Points in Mathematics Functions
Analyzing Local Maxima and Minima | Insights into Function Behavior and Critical Features