Problem

Find f such that f(x)=10x3,f(2)=0
f(x)=

Answer

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Answer

f(x)=5x23x14 is the function that satisfies the given conditions.

Steps

Step 1 :The problem is asking for a function f(x) such that its derivative f(x) is equal to 10x3 and f(2) is equal to 0. This is a problem of finding the antiderivative (or integral) of a function, and then adjusting the constant of integration such that the function passes through the point (2, 0).

Step 2 :The antiderivative of 10x3 is 5x23x+C, where C is the constant of integration.

Step 3 :We can find the value of C by substituting x=2 and f(2)=0 into the equation and solving for C.

Step 4 :Substituting these values into the equation gives us C+14=0, so C=14.

Step 5 :Substituting C=14 back into the equation gives us the function f(x)=5x23x14.

Step 6 :f(x)=5x23x14 is the function that satisfies the given conditions.

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