Find the function whose tangent has slope 𝑥 3 − 2 𝑥 2 + 16 for each of x and whose graph passes through the point (2,5).

The function whose tangent has a given slope is the integral of that slope. So, we first need to find the integral of the given function, x^3 - 2x^2 + 16.

∫(x^3 - 2x^2 + 16) dx = (1/4)x^4 - (2/3)x^3 + 16x + C

Now, we know that the graph of the function passes through the point (2,5). We can use this information to find the value of C.

5 = (1/4)(2)^4 - (2/3)(2)^3 + 16(2) + C 5 = 4 - 16/3 + 32 + C 5 = 36/3 - 16/3 + 96/3 + C 5 = 116/3 + C 5 - 116/3 = C C = -86/3

So, the function is f(x) = (1/4)x^4 - (2/3)x^3 + 16x - 86/3.