Question

Find f, f ″(x) = 12x^3 + 54x − 1 (use c for constant of first derivative and d for constant of second derivative)

Find f. f ''(x) = 8 + 6x + 24x2, f(0) = 3, f (1) = 15

Answer 1

The antiderivative f(x) of the second derivative f''(x) = 12x³ + 54x - 1 is found to be f(x) = 2x⁴ + 27x² - x + c, where c is the constant of integration. To determine the specific function f(x) and the constant c, we need additional information. Utilizing the given initial conditions f(0) = 3 and f(1) = 15, we can solve for c, resulting in the final function f(x).

Step-by-step explanation:

To find the antiderivative f(x of the second derivative f''(x), we integrate each term of f''(x) = 12x³ + 54x - 1 with respect to x. The antiderivative is obtained as f(x) = 2x⁴ + 27x² - x + c, where c is the constant of integration.

To determine the specific function f(x) and the value of c, we apply the initial conditions provided: f(0) = 3 and f(1) = 15. Plugging x = 0 into f(x) and equating it to 3 allows us to solve for c. Subsequently, c is used to find the complete function f(x).

Understanding the process of finding antiderivatives and integrating polynomial functions is fundamental in calculus. The constant of integration c accounts for the family of antiderivatives, and specific values are determined by applying initial conditions or constraints provided in the problem.

Answer 2

`\int\left(12 x^3+54 x-1\right) d x=3 x^4+27 x^2-x+C`

We can find the indefinite integral of the expression using the power rule of integration. This rule states that the integral of
`$x^n$` is
`(x^(n+1))/(n+1)`, where
`$n$` is any real number except for -1 .

Apply the power rule of integration:

`\int\left(12 x^3+54 x-1\right) d x=12 \cdot (x^(3+1))/(3+1)+54 \cdot (x^(1+1))/(1+1)-(x^(0+1))/(0+1)+C`

Simplify the expression:

`12 \cdot (x^4)/(4)+54 \cdot (x^2)/(2)-(x^1)/(1)+C`

Combine terms:

`3 x^4+27 x^2-x+C`

Answer

`3 x^4+27 x^2-x+C`

The constant of integration,
`$C$`, is added at the end because the derivative of any constant is zero.