Which of the following functions is differentiable at x = 1? f(x) = { x^2, x <
Practice Questions
Q1
Which of the following functions is differentiable at x = 1? f(x) = { x^2, x < 1; 2x - 1, x >= 1 }
f(1) = 1
f(1) = 0
f(1) = 2
f(1) = 3
Questions & Step-by-Step Solutions
Which of the following functions is differentiable at x = 1? f(x) = { x^2, x < 1; 2x - 1, x >= 1 }
Step 1: Identify the function pieces. The function f(x) is defined as f(x) = x^2 for x < 1 and f(x) = 2x - 1 for x >= 1.
Step 2: Check the value of f(1). Since 1 is in the second piece, calculate f(1) = 2(1) - 1 = 1.
Step 3: Check the left-hand limit as x approaches 1. Calculate the limit of f(x) as x approaches 1 from the left: lim (x -> 1-) f(x) = lim (x -> 1-) x^2 = 1.
Step 4: Check the right-hand limit as x approaches 1. Calculate the limit of f(x) as x approaches 1 from the right: lim (x -> 1+) f(x) = lim (x -> 1+) (2x - 1) = 1.
Step 5: Since the left-hand limit and right-hand limit both equal f(1), the function is continuous at x = 1.
Step 6: Calculate the left-hand derivative at x = 1. The derivative of f(x) = x^2 is f'(x) = 2x, so f'(1) = 2(1) = 2.
Step 7: Calculate the right-hand derivative at x = 1. The derivative of f(x) = 2x - 1 is f'(x) = 2, so f'(1) = 2.
Step 8: Compare the left-hand and right-hand derivatives. Since both derivatives equal 2, the function is differentiable at x = 1.
