Answer:
Step-by-step explanation:
Here we have the second derivative of a function and we need to go back to the original function. This situation is inverse of taking derivative which is called antiderivative or integral. But because we have the second derivative, we have to integrate twice because the first integral will give me the first derivative then the second integral will give me the function itself.
1. Let's start taking the 1st integral of x-2 which is x²/2-2x+C1
2. Take the 2nd integral which is the integral of the 1st one
x³/6-2x²/2+C1x+C2
remember: the integral is the inverse of derivative, that means we have to add 1 to the power of x and divide by that power. I did that in steps 1 and 2.
3. After simplifying the second term, f(x)=x³/6-x²+C1x+C2 this is the function, but I have to find C1 and C2 the constants. We can use the two conditions given. the 1st one means that when x=1 , y=0
0=1/6-1+C1+C2 now combine like terms and move them to the left side of the equation C1+C2=5/6
the 2nd condition means that when x=3 , y=0
0=27/6-9+3C1+C2 after simplifying 3C1+C2=9/2
4. We got 2 equations with 2 variables C1 and C2, we can solve these system of equations. You can eliminate C2 and find C1 which is 11/6 then plug in C1 in any of the two equations to find C2, which is -1
5. Last step substitute C1 and C2 in the function from step 3 and you will get f(x)=x³/6-x²+11x/6-1