Question

find all points \\(x\\) where the function \\(g(x) = |x + 5| - |x - 9|\\) is not differentiable. separate multiple answers with a comma. if there are no points where the function is not differentiable, write none for your answer.

\\(g(x)\\) is not differentiable at \\(x =\\)

Explanation:

Identify critical points of absolute value terms

The function is defined as:

$$ g(x) = |x + 5| - |x - 9| $$

The individual absolute value terms \(|x + 5|\) and \(|x - 9|\) have sharp corners (cusps) where their arguments equal zero:

$$ x + 5 = 0 \implies x = -5 $$

$$ x - 9 = 0 \implies x = 9 $$

Analyze differentiability at critical points

For \(x = -5\):
The term \(|x - 9|\) is differentiable near \(x = -5\) because its argument is non-zero.
The term \(|x + 5|\) has a corner at \(x = -5\), where the left-hand derivative is \(-1\) and the right-hand derivative is \(1\).
Thus, \(g(x)\) is not differentiable at \(x = -5\).

For \(x = 9\):
The term \(|x + 5|\) is differentiable near \(x = 9\) because its argument is non-zero.
The term \(|x - 9|\) has a corner at \(x = 9\), where the left-hand derivative is \(-1\) and the right-hand derivative is \(1\).
Thus, \(g(x)\) is not differentiable at \(x = 9\).

At all other points \(x
eq -5\) and \(x
eq 9\), both absolute value terms are differentiable, making \(g(x)\) differentiable.

Formulate the final answer

The points where \(g(x)\) is not differentiable are \(x = -5\) and \(x = 9\).
We separate multiple answers with a comma: \(-5, 9\).

Answer:

\(g(x)\) is not differentiable at \(x =\) <blank>\(-5, 9\)</blank>