Find the most general antiderivative of the function. (Check your answer by differentiation. Use C for the constant of the antiderivative.)f(x) = 6x5 − 7x4 − 9x2F(x) =

To find the most general antiderivative of the function f(x) = 6x^5 - 7x^4 - 9x^2, we need to apply the power rule for antiderivatives, which states that the antiderivative of x^n is (1/(n+1))x^(n+1).

Step 1: Apply the power rule to each term in the function.

For the first term, 6x^5, the antiderivative is (1/(5+1))6x^(5+1) = x^6.

For the second term, -7x^4, the antiderivative is (1/(4+1))(-7)x^(4+1) = -7/5 * x^5.

For the third term, -9x^2, the antiderivative is (1/(2+1))(-9)x^(2+1) = -3x^3.

Step 2: Combine the antiderivatives of each term to find the most general antiderivative of the function.

F(x) = x^6 - 7/5 * x^5 - 3x^3 + C

where C is the constant of the antiderivative.

Step 3: Check the answer by differentiation.

The derivative of F(x) = x^6 - 7/5 * x^5 - 3x^3 + C is f(x) = 6x^5 - 7x^4 - 9x^2, which is the original function, so the antiderivative is correct.