If given function, (shown in the image) is differentiable then find |a+p|

To determine the value of |a + p| for a given differentiable function, we need to analyze the properties of differentiability and the implications it has on the function's behavior. Differentiability at a point means that the function must be continuous at that point, and its derivative must exist. This often involves examining the function's limits and ensuring that they match at the point of interest.

Understanding Differentiability

A function is said to be differentiable at a point if the derivative exists at that point. This requires that the left-hand limit and the right-hand limit of the derivative must be equal. For a function defined piecewise or with specific parameters, we often need to ensure that the function is continuous and smooth at the transition points.

Analyzing the Function

Let’s assume the function is defined in a piecewise manner, where different expressions apply to different intervals. To find |a + p|, we typically need to set up equations based on the conditions of differentiability. Here’s how we can approach it:

• Step 1: Identify the points where the function changes its definition. These points are crucial for checking continuity and differentiability.
• Step 2: Ensure that the function is continuous at these points. This means that the left-hand limit and right-hand limit must equal the function value at that point.
• Step 3: Calculate the derivatives from both sides of the transition points. Set these derivatives equal to each other to ensure differentiability.

Example Scenario

Suppose we have a function defined as follows:

f(x) = { ax + b for x < c; px + d for x ≥ c }

To find |a + p|, we would:

• Set the two expressions equal at x = c to ensure continuity: ac + b = pc + d.
• Calculate the derivatives: f'(x) = a for x < c and f'(x) = p for x ≥ c.
• Set the derivatives equal: a = p.

Solving for |a + p|

From the condition a = p, we can substitute p into our continuity equation. This gives us:

ac + b = ac + d, which simplifies to b = d.

Now, since we have established that a = p, we can express |a + p| as |a + a| = |2a|. Therefore, the final answer will depend on the specific value of a.

Final Thoughts

In summary, to find |a + p|, we need to ensure that the function is both continuous and differentiable at the point where its definition changes. By setting up the necessary equations and solving them, we can derive the relationship between a and p, leading us to the desired absolute value. If you have specific values for a and p, we can plug those in to find the exact numerical result.