Find the value of k for which the function f(x) = kx^2 + 3x + 2 is differentiabl

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Question: Find the value of k for which the function f(x) = kx^2 + 3x + 2 is differentiable everywhere.

Options:

Correct Answer: k = 0

Solution:

The function is a polynomial and is differentiable for all k, hence k can be any real number.

Find the value of k for which the function f(x) = kx^2 + 3x + 2 is differentiable everywhere.

Find the value of k for which the function f(x) = kx^2 + 3x + 2 is differentiable everywhere.

Correct Answer: Any real number

Step 1: Understand what it means for a function to be differentiable. A function is differentiable everywhere if it has a derivative at every point in its domain.
Step 2: Identify the type of function given. The function f(x) = kx^2 + 3x + 2 is a polynomial function.
Step 3: Recall the properties of polynomial functions. Polynomial functions are smooth and continuous, which means they are differentiable everywhere.
Step 4: Conclude that since f(x) is a polynomial, it does not depend on the value of k for differentiability.
Step 5: State that k can be any real number because the function will still be differentiable for all values of k.

Differentiability of Polynomials – Polynomials are differentiable everywhere on the real number line, meaning they do not have any points of non-differentiability.