Answer :
Final answer:
The x-coordinates of the relative maxima of the function can be found by taking the first and second derivatives, setting them equal to zero, and solving for x. Then, we verify which values represent a relative maximum by checking if the second derivative is negative. This represents the usual process for finding relative maxima in calculus.
Explanation:
To find the x-coordinates of the relative maxima of the given function f(x) = -(3/5)x⁵+9x⁴-35x³+6, we first take the derivative of the function, set it to zero, and solve for x. This procedure will give us the critical points of the function.
The derivative, f'(x), of the function is -3x⁴+36x³-105x². Setting this equal to zero gives the equation -3x⁴+36x³-105x²=0. This equation can be solved by factoring out an x², yielding -x²(3x²-36x+105)=0. Next, setting each factor to zero, we obtain x=0 and the roots of the quadratic equation 3x²-36x+105=0. The x that satisfies the quadratic equation will be the x-coordinates of the relative maxima. However, we need to double-check by taking the second derivative and substitute the x-values into it. A positive result indicates a minimum so we disregard those, and a negative result indicates a maximum.
Remember the process: first derivative, second derivative, and checking the values on the second derivative test is the common approach for finding relative maxima.
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