Question

Find \(f\)

\(f'\left( x \right) = 5{x^4} - 3{x^2} + 4,\;f\left( { - 1} \right) = 2\)

Short answer

Required function is \(f\left( x \right) = {x^5} - {x^3} + 4x + 6\).

Step-by-step solution

Find a general antiderivative of \(f'\)

Here we have,

\(f'\left( x \right) = 5{x^4} - 3{x^2} + 4\)

Now, the general antiderivative of \(f'\left( x \right) = 5{x^4} - 3{x^2} + 4\) is,

\(\begin{aligned}{c}f\left( x \right) &= 5\frac{{{x^5}}}{5} - 3\frac{{{x^3}}}{3} + 4x + C\\ &= {x^5} - {x^3} + 4x + C\end{aligned}\)

Find the value of constant \(C\)

Now, we have, \(f\left( x \right) = {x^5} - {x^3} + 4x + C\)

Also, we have \(f\left( { - 1} \right) = 2\)

Now, by putting \(f\left( { - 1} \right) = 2\) in \(f\left( x \right) = {x^5} - {x^3} + 4x + C\) we get,

\(\begin{aligned}{l} \Rightarrow 2 &= {\left( { - 1} \right)^5} - {\left( { - 1} \right)^3} + 4\left( { - 1} \right) + C\\ \Rightarrow 2 &= - 1 + 1 - 4 + C\\ \Rightarrow C &= 6\end{aligned}\)

So, required function is \(f\left( x \right) = {x^5} - {x^3} + 4x + 6\).