GATE2025_CS1_Q31

Consider the given function f(x).

f(x) = { ax + b,   for x < 1
x³ + x² + 1,   for x ≥ 1

If the function is differentiable everywhere, the value of b must be ______. (rounded off to one decimal place)

✅ Final Answer:

The correct answer is -2.0.

🔹 Short Answer:

For the function to be differentiable at the critical point x=1, it must be both continuous and have equal left-hand and right-hand derivatives. Continuity at x=1 gives the equation a + b = 3. Differentiability at x=1 gives a = 5. Substituting a = 5 into the first equation gives 5 + b = 3, which yields b = -2. Rounded to one decimal place, the answer is -2.0.

🔸 Long Answer:

For a piecewise function to be differentiable everywhere, it must satisfy two conditions at the point where the function's definition changes (in this case, at x = 1):

1. The function must be continuous at that point.
2. The function must be "smooth" at that point, meaning the derivative from the left must equal the derivative from the right.

Step 1: Apply the Continuity Condition at x = 1

For f(x) to be continuous at x = 1, the limit as x approaches 1 from the left must equal the limit as x approaches 1 from the right.

lim_x→1⁻ f(x) = lim_x→1⁺ f(x)

Substitute the respective function pieces:

lim_x→1⁻ (ax + b) = lim_x→1⁺ (x³ + x² + 1)

Now, evaluate the limits by plugging in x = 1:

a(1) + b = (1)³ + (1)² + 1
a + b = 1 + 1 + 1
a + b = 3 --- (Equation 1)

Step 2: Apply the Differentiability Condition at x = 1

First, we find the derivative, f'(x), for each piece of the function.

• For x < 1:   f'(x) = d/dx (ax + b) = a
• For x > 1:   f'(x) = d/dx (x³ + x² + 1) = 3x² + 2x

For the function to be differentiable at x = 1, the left-hand derivative (LHD) must equal the right-hand derivative (RHD) at x = 1.

f'_-(1) = f'₊(1)

Evaluate each derivative at x = 1:

a = 3(1)² + 2(1)
a = 3 + 2
a = 5 --- (Equation 2)

Step 3: Solve for b

Now we have a simple system of two equations. Substitute the value of a from Equation 2 into Equation 1:

a + b = 3
(5) + b = 3
b = 3 - 5
b = -2

Step 4: Format the Final Answer

The question requires the answer to be rounded off to one decimal place. Therefore, -2 becomes -2.0.

Bibliography:
- Thomas, G. B., Weir, M. D., & Hass, J. (2014). Thomas' Calculus. Pearson. (Chapter 3: Differentiation).

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