Question:
If for a continuous function f, f(0) = f(1) = 0, f'(1) = 2 and \(g(x) = f(e^x) e^{f(x)}\), then g'(0) is equal to
यदि एक संतत फलन f के लिए, f(0) = f(1) = 0, f'(1) = 2 और \(g(x) = f(e^x) e^{f(x)}\) है, तो g'(0) किसके बराबर है?
Options:
- 3
- 0
- 2
- 1
- 3
- 0
- 2
- 1
Correct Answer: c)
Explanation:
\(g'(x) = f'(e^x)e^x e^{f(x)} + f(e^x)e^{f(x)}f'(x)\).
\(g'(0) = f'(1)e^0 e^{f(0)} + f(1)e^{f(0)}f'(0) = 2(1)(1) + 0 = 2\).
\(g'(0) = f'(1)e^0 e^{f(0)} + f(1)e^{f(0)}f'(0) = 2(1) + 0 = 2\)।