Question

Find the function f(x) described by the given initial value problem.
f′′(x)=0, f′(1)=3, f(1)=1
f(x)= ?

Answer 1

SOLUTION

TO DETERMINE

The function f(x) described by the given initial value problem

f′′(x) = 0 , f′(1) = 3 , f(1) = 1

EVALUATION

Here it is given that

f′′(x) = 0

On integration we get

f′(x) = a - - - - - (1)

Now f′(1) = 3 gives

f′(1) = a

$\implies \sf{3 = a}$

$\implies \sf{a = 3}$

From Equation 1 we get

f′(x) = 3 - - - - - (2)

Again on integration we get

f(x) = 3x + b

Now f(1) = 1 gives

1 = 3 + b

$\implies \sf{3 + b = 1}$

$\implies \sf{b = - 2}$

This gives

f(x) = 3x - 2

FINAL ANSWER

Hence the required function f(x) = 3x - 2

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