Applications of Derivatives
Self-Referential Polynomial — Finding Derivative Values
nta_pyq_2024_jan
Grade 12
Question:
Let $f(x)=x^3+x^2f'(1)+xf''(2)+f'''(3)$, $x\in\mathbb{R}$. Then $f'(10)$ is equal to
Step-by-Step Solution
Key Concept: Let $f'(1)=a$, $f''(2)=b$, $f'''(3)=c$. Differentiate $f(x)$ repeatedly to get expressions for $a,b,c$. Solve the system and then compute $f'(10)$.
$f'''=6\Rightarrow c=6$. $f''(x)=6x+2a\Rightarrow b=12+2a$. $f'(x)=3x^2+2ax+b\Rightarrow a=3+2a+b\Rightarrow a=-5,b=2$. $f'(x)=3x^2-10x+2$. $f'(10)=202$.
Correct Answer: 202