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Answer :
To find the coordinates of all relative maxima for the function [tex]\( f(x) = \frac{3}{5} x^5 + 12 x^4 + 60 x^3 + 3 \)[/tex], we need to follow these steps:
1. Find the First Derivative:
The first step is to find the derivative of the function, which is necessary to locate the critical points. Critical points occur where the first derivative is zero or undefined, indicating possible maximum or minimum points.
2. Set the First Derivative to Zero:
Solve the equation obtained by setting the first derivative to zero. This will give us the x-values of the critical points.
3. Evaluate the Second Derivative:
Compute the second derivative of the original function. The second derivative test helps us determine the nature of the critical points. If the second derivative is negative at a critical point, the function has a relative maximum at that point.
4. Apply the Second Derivative Test:
Substitute each critical point into the second derivative. If the result is negative, the function has a relative maximum at that critical point.
5. Calculate the Function Value at the Relative Maximum:
Finally, for each critical point determined to be a maximum, plug the x-value back into the original function to find the corresponding y-value.
Following these steps, the critical point at which a relative maximum occurs is [tex]\( x = -10 \)[/tex]. When substituted back into the function, it yields the function value of 3. Therefore, the coordinates of the relative maximum are [tex]\((-10, 3)\)[/tex].
1. Find the First Derivative:
The first step is to find the derivative of the function, which is necessary to locate the critical points. Critical points occur where the first derivative is zero or undefined, indicating possible maximum or minimum points.
2. Set the First Derivative to Zero:
Solve the equation obtained by setting the first derivative to zero. This will give us the x-values of the critical points.
3. Evaluate the Second Derivative:
Compute the second derivative of the original function. The second derivative test helps us determine the nature of the critical points. If the second derivative is negative at a critical point, the function has a relative maximum at that point.
4. Apply the Second Derivative Test:
Substitute each critical point into the second derivative. If the result is negative, the function has a relative maximum at that critical point.
5. Calculate the Function Value at the Relative Maximum:
Finally, for each critical point determined to be a maximum, plug the x-value back into the original function to find the corresponding y-value.
Following these steps, the critical point at which a relative maximum occurs is [tex]\( x = -10 \)[/tex]. When substituted back into the function, it yields the function value of 3. Therefore, the coordinates of the relative maximum are [tex]\((-10, 3)\)[/tex].
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