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

NTA Abhyas 2020 · Mathematics · Differentiation

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

Let f(x)f \left(x\right) be a differentiable function on x∈Rx \in \mathbb{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,yx, y. If f(0)≠0,f(5)=12f \left(0\right) \ne 0, f \left(5\right) = 12 and f′(0)=16f^{′} \left(0\right) = 16, then f′(5)isequaltof^{′} \left(5\right) is equal to

Options

  1. A190190
  2. B186186
  3. C196196
  4. D192192

Correct answer

D. 192192

Step-by-step solution

f′(5)=lim⁡h→0f(5+h)−f(5)hf^{′} \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) \cdot f\left(h\right) - f\left(5\right) \cdot f \left(0\right)}{h}
\left[∵∵f\left(x + y\right) = f\left(x\right) . f\left(y\right)forallx, y\right]
=(lim⁡h→0f(h)−f(0)h)\cdotf(5)= \left(\lim_{h \to 0}\dfrac{f \left( h\right) - f\left(0\right)}{h}\right)\cdotf\left(5\right)
f′(0)\timesf(5)=16×12=192f^{′} \left(0\right) \timesf\left(5\right) = 16 \times12 = 192
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