NIMCET 2012 — Mathematics PYQ
NIMCET | Mathematics | 2012If f(a+b)=f(a)×f(b) for all a and b and f(5)=2,f′(0)=3, then f′(5) is equal to
Choose the correct answer:
- A.
2
- B.
4
- C.
6
(Correct Answer) - D.
8
6
Explanation
1. Determine f(0)
Using the functional equation f(a+b)=f(a)⋅f(b), let a=5 and b=0:
Since f(5)=2 (which is non-zero), we can divide both sides by f(5):
2. Use the definition of f′(x)
The derivative at any point x is defined as:
Using the given property f(x+h)=f(x)⋅f(h):
3. Relate to f′(0)
We know that f′(0) is defined as:
Since f(0)=1:
From the problem, we are given f′(0)=3.
4. Calculate f′(5)
Substituting the limit value back into our general expression for f′(x):
For x=5:
Substitute the given values f(5)=2 and f′(0)=3:
Correct Option:
(c) 6
Explanation
1. Determine f(0)
Using the functional equation f(a+b)=f(a)⋅f(b), let a=5 and b=0:
Since f(5)=2 (which is non-zero), we can divide both sides by f(5):
2. Use the definition of f′(x)
The derivative at any point x is defined as:
Using the given property f(x+h)=f(x)⋅f(h):
3. Relate to f′(0)
We know that f′(0) is defined as:
Since f(0)=1:
From the problem, we are given f′(0)=3.
4. Calculate f′(5)
Substituting the limit value back into our general expression for f′(x):
For x=5:
Substitute the given values f(5)=2 and f′(0)=3:
Correct Option:
(c) 6