Find f if f''(x)=24x^2+2x+10, f(1)=-5, f'(1)=-7. f(x)=?

Answer 1

Given: f'' = 24x^2 + 2x + 10

With: f'(1) = 7

Find: f(x)

f' = Integral of f''

f' = Integral [ 24x^2 + 2x + 10 ]

f' = (24/3)x^3 + (2/2)x^2 + 10x + C

f' = 8x^3 + x^2 + 10x + C

Use; f'(1) = 7 to find C

f' = 8x^3 + x^2 + 10x + C

7 = 8(1)^3 + (1)^2 + 10(1) + C

7 = 8 + 1 + 10 + C

7 = 19 + C

-12 = C

Therefore:

f'(x) = 8x^3 + x^2 + 10x + C

f'(x) = 8x^3 + x^2 + 10x - 12

Answer 2

Since this seems to be a homework problem, I'll give a more general answer.

You are given the second derivative of a function, the value of the function at one point, and the value of the derivative of the function at another point, and asked to find the function.

To do so, you will integrate the second derivative of the function twice, and use the initial values to solve for the constants of integration. As an example:

Find f if f''(x) = 2x + 2, f'(0) = 1, f(1) = 2:

∫ 2x + 2 = x² + 2x + C = f'(x)

To find the value of C, we know that f'(0) = 1 = C (plugging in x = 0 in the expression we found)

Now we integrate f'(x) = x² + 2x + 1:

∫ x² + 2x + 1 = (1/3)x³ + x² + x + C.

We then find C using our initial value, f(1) = 2: this gives us C = - 1/3. So our final answer is f(x) = (1/3)x³ + x² + x -1/3.