For which value of a is the function f(x) = x^2 + ax + 1 differentiable everywhe

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Question: For which value of a is the function f(x) = x^2 + ax + 1 differentiable everywhere?

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Correct Answer: 0

Solution:

The function is a polynomial and is differentiable for all real numbers, hence any value of a works.

For which value of a is the function f(x) = x^2 + ax + 1 differentiable everywhere?

For which value of a is the function f(x) = x^2 + ax + 1 differentiable everywhere?

Step 1: Understand what it means for a function to be differentiable. A function is differentiable everywhere if you can find its derivative at every point in its domain.
Step 2: Identify the type of function given. The function f(x) = x^2 + ax + 1 is a polynomial function.
Step 3: Know that polynomial functions are smooth and continuous. This means they do not have any breaks, jumps, or sharp corners.
Step 4: Realize that all polynomial functions are differentiable everywhere on the real number line, regardless of the coefficients (like 'a' in this case).
Step 5: Conclude that since f(x) is a polynomial, it is differentiable for all values of 'a'.

Differentiability of Polynomials – Polynomials are differentiable everywhere on the real number line, meaning they have a derivative at every point.