f ''(x) = 10 + 6x + 24x2,    f(0) = 4,    f (1) = 16

The given function is the second derivative of the original function f(x). To find the original function, we need to integrate twice and use the given points to find the constants of integration.

Step 1: Integrate f''(x) to find f'(x)

∫f''(x) dx = ∫(10 + 6x + 24x^2) dx = 10x + 3x^2 + 8x^3 + C1

Step 2: Integrate f'(x) to find f(x)

∫f'(x) dx = ∫(10x + 3x^2 + 8x^3 + C1) dx = 5x^2 + x^3 + 2x^4 + C1x + C2

Now, we use the given points to find the constants C1 and C2.

Step 3: Use f(0) = 4 to find C2

Substitute x = 0 into f(x) = 5x^2 + x^3 + 2x^4 + C1x + C2, we get C2 = 4.

Step 4: Use f(1) = 16 to find C1

Substitute x = 1 into f(x) = 5x^2 + x^3 + 2x^4 + C1x + C2, we get 5 + 1 + 2 + C1 + 4 = 16, which simplifies to C1 = 4.

So, the original function is f(x) = 5x^2 + x^3 + 2x^4 + 4x + 4.