Answer :
not continuous: x = -5. not differentiable: x = -5.
The absolute value function itself (f(x) = |x|) is continuous for all real numbers. However, when the absolute value function is combined with other functions, there might be points where continuity or differentiability is lost. Let's analyze the given function:
f(x) = 7|x + 5|
Continuity:
In our function f(x) = 7|x + 5|, the absolute value creates a V-shaped corner point at x = -5. As x approaches -5 from either side, the function approaches different values due to the absolute value's behavior. This violates the definition of continuity, which requires the function's value to approach the limit as x approaches a specific point.
The absolute value function f(x) = 7|x + 5| is not continuous at the point: x = -5.
Differentiability:
- The absolute value function is not differentiable at the point where it changes direction. This happens at x = 0 because the absolute value function transitions from a negative slope (-1 for x < 0) to a positive slope (1 for x > 0) at x = 0.
- In our function, f(x) = 7|x + 5|, the absolute value is taken within the expression (x + 5). This shift to the left by 5 units doesn't affect the "change in direction" property of the absolute value function. Therefore, the function f(x) = 7|x + 5| will also lose differentiability at the point where the absolute value function itself is not differentiable, which is:
x = -5 (because the absolute value function transitions at x = 0, and here it's shifted 5 units to the left).