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NTA Abhyas 2020
Exam
NTA Abhyas
Year
2020
Subject
Mathematics
Chapter
Differentiation

NTA Abhyas 2020 · Mathematics · Differentiation

Let f(x) is a differentiable function on x ∈ R , such that f(x + y)...

Let f(x)f\left(x\right) is a differentiable function on x∈Rx \in R, such that f(x+y)=f(x)f(y)f\left(x + y\right) = f\left(x\right)f\left(y\right) for all x,y∈Rx, y \in R where f(0)≠0.f\left(0\right) \ne 0 . If f(5)=10,f′(0)=6f\left(5\right) = 10, {f}^{'}\left(0\right) = 6, then the value of f′(5){f}^{′} \left(5\right) is equal to

Correct answer

60

Step-by-step solution

f′(5)=lim⁡h→0f(5+h)−f(5)h{f}^{′} \left(5\right) = \lim_{h \to 0} \dfrac{f\left(5 + h\right) - f\left(5\right)}{h}
=lim⁡h→0f(5+h)−f(5+0)h= \lim_{h \to 0}\dfrac{f\left(5 + h\right) - f\left(5 + 0\right)}{h}
=lim⁡h→0f(5).f(h)−f(5).f(0)h= \lim_{h \to 0}\dfrac{f\left(5\right) . f\left(h\right) - f\left(5\right) . f\left(0\right)}{h}
[∵f(x+y)=f(x).f(y)forallx,y][ ∵ f\left(x + y\right) = f\left(x\right) . f\left(y\right)forallx, y ]
=(lim⁡h→0f(h)−f(0)h).f(5)= \left(\lim_{h \to 0}\dfrac{f\left(h\right) - f\left(0\right)}{h}\right) . f\left(5\right)
=f′(0).f(5)=6×10= {f}^{′} \left(0\right) . f\left(5\right) = 6 \times 10
=60= 60
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