If $f(x) = {x^3} - {x^2}f'(1) + xf''(2) - f'''(3),x \in \mathbb{R}$, then
Solution
$$
f(x)=x^3-x^2 f^{\prime}(1)+x f^{\prime \prime}(2)-f^{\prime \prime \prime}(3), x \in R
$$<br/><br/>
Let $\mathrm{f}^{\prime}(1)=\mathrm{a}, \mathrm{f}^{\prime \prime}(2)=\mathrm{b}, \mathrm{f}^{\prime \prime \prime}(3)=\mathrm{c}$<br/><br/>
$$
\begin{aligned}
& f(x)=x^3-a x^2+b x-c \\\\
& f^{\prime}(x)=3 x^2-2 a x+b \\\\
& f^{\prime \prime}(x)=6 x-2 a \\\\
& f^{\prime \prime \prime}(x)=6 \\\\
& c=6, a=3, b=6 \\\\
& f(x)=x^3-3 x^2+6 x-6 \\\\
& f(1)=-2, f(2)=2, f(3)=12, f(0)=-6 \\\\
& 2 f(0)-f(1)+f(3)=2=f(2)
\end{aligned}
$$
About this question
Subject: Mathematics · Chapter: Differentiation · Topic: Derivatives of Standard Functions
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