Question

Assertion: For the function
$f ( x )=\frac{ x ^{100}}{100}+\frac{ x ^{99}}{99}+\ldots+\frac{ x ^{2}}{2}+ x +1, f '(1)=100 f '(0)$
Reason: $\frac{ d }{ dx }\left( x ^{ n }\right)= n \cdot x ^{ n -1}$.

• A. Assertion is correct, reason is correct; reason is a correct explanation for assertion
• B. Assertion is correct, reason is correct; reason is not a correct explanation for assertion
• C. Assertion is correct, reason is incorrect
• D. Assertion is incorrect, reason is correct

Answer 1

Answer: (A)

We know that $\frac{d}{d x}\left(x^{n}\right)=n x^{n-1}$
$\therefore $ For $f(x)=\frac{x^{100}}{100}+\frac{x^{99}}{99}+\ldots . .+\frac{x^{2}}{2}+x+1$
$f'(x)=\frac{100 x^{99}}{100}+99 \frac{x^{98}}{99}+\ldots . .+\frac{2 x}{2}+1$
$=x^{99}+x^{98}+\ldots \ldots \ldots+x+1$
Now, $f'(1)=1+1+\ldots \ldots \ldots .$ to 100 term $=100$
$f'(0)=1 \therefore f^{\prime}(1)=100 \times 1=100 f'(0)$
Hence, $f'(1)=100 f'(0)$