Consider the following statements : 1. f(x) is differentiable for <strong>ALL</strong> x \lt 0 2. g(x) is continuous at x=0.0001 3. The derivative of g(x) at x=2.5 is 1 Which of the statements given above are correct?
Consider the following for the next two (02) items that follow : Let f(x)=|x|+1 and g(x)=[x]-1, where [.] is the greatest integer function. Let h(x)=\frac{f(x)}{g(x)}
- A. 1 and 2 only ✓
- B. 2 and 3 only
- C. 1 and 3 only
- D. 1, 2 and 3
Correct Answer: A. 1 and 2 only
Explanation
For x \lt 0, f(x) = -x+1, which is a straight line and differentiable everywhere on its domain, so 1 is true. Near x=0.0001, [x]=0, meaning g(x) = -1 (a constant), which is continuous, so 2 is true. At x=2.5, [x] is locally constant, so g(x) = 1. The derivative of a constant is 0, not 1, making statement 3 false.
Related questions on Calculus
- Let z=[y] and y=[x]-x, where [.] is the greatest integer function. If x is <strong>NOT</strong> an integer but positive, then what i...
- If f(x)=4x+1 and g(x)=kx+2 such that fog(x)=gof(x), then what is the value of k?
- What is the <strong>MINIMUM</strong> value of the function f(x)=\log_{10}(x^{2}+2x+11)?
- What is \int(x^{x})^{2}(1+\ln x)\,dx equal to ?
- What is \int e^{x}\{1+\ln x+x\ln x\}\,dx equal to?