AP EAMCET · Maths · Continuity and Differentiability
Let \(f(x)\) be a real valued function. If \(f^{\prime}(x)\) is a constant for all \(x \in \mathbb{R}, f(0)=2\) and \(f^{\prime}(0)=1\), then
- A \(\mathrm{f}(\mathrm{x})\) is not continuous on \(\mathbb{R}\)
- B \(f(x)\) is continuous at \(x=0,1,2\) and 3 only
- C \(f(x)\) is continuous only on \([0, \infty)\)
- D \(\mathrm{f}(\mathrm{x})\) is continuous on \(\mathbb{R}\)
Answer & Solution
Correct Answer
(D) \(\mathrm{f}(\mathrm{x})\) is continuous on \(\mathbb{R}\)
Step-by-step Solution
Detailed explanation
Since \(f^{\prime}(x)\) is a constant \[ \therefore f^{\prime}(x)=a \text { (say) }...(1) \] \(\Rightarrow f(x)=a x+b\) where \(b\) is arbitrary constant. ...(2) Since \(f(0)=2 \Rightarrow b=2\) Since \(f^{\prime}(0)=1 \Rightarrow a=1\) \[ \therefore f(x)=(x+2) \] Which is…
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