The equation of the curve is `y=f(x)dot` The tangents at `[1,f(1),[2,f(2)],a n d[3,f(3)]` make angles `pi/6,pi/3,a n dpi/4,` respectively, with the positive direction of x-axis. Then the value of `int_2^3f^(prime)(x)f^('')dx+int_1^3f^('')dx` is equal to `-1/(sqrt(3))` (b) `1/(sqrt(3))` (e) 0 (d) none of these

Options

A. `-1//sqrt(3)`
B. `1//sqrt(3)`
C. `0`
D. none of these

Explanation

To solve the problem step-by-step, we will follow the reasoning laid out in the video transcript.

### Step 1: Identify the derivatives from the angles
Given the angles made by the tangents at points (1, f(1)), (2, f(2)), and (3, f(3)), we can find the derivatives at these points using the tangent of the angles:

1. \( f'(1) = \tan\left(\frac{\pi}{6}\right) = \frac{1}{\sqrt{3}} \)
2. \( f'(2) = \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \)
3. \( f'(3) = \tan\left(\frac{\pi}{4}\right) = 1 \)

### Step 2: Set up the integrals
We need to evaluate the expression:

\[ \int_2^3 f'(x) f''(x) \, dx + \int_1^3 f''(x) \, dx \]

### Step 3: Evaluate the first integral
Using integration by parts on the first integral:

Let \( u = f'(x) \) and \( dv = f''(x) \, dx \).

Then, \( du = f''(x) \, dx \) and \( v = f'(x) \).

The integration by parts formula is:

\[ \int u \, dv = uv - \int v \, du \]

Applying this, we have:

\[ \int_2^3 f'(x) f''(x) \, dx = \left[ f'(x)^2 / 2 \right]_2^3 \]

Calculating this gives:

\[ \frac{f'(3)^2}{2} - \frac{f'(2)^2}{2} = \frac{1^2}{2} - \frac{(\sqrt{3})^2}{2} = \frac{1}{2} - \frac{3}{2} = -1 \]

### Step 4: Evaluate the second integral
Now, we evaluate the second integral:

\[ \int_1^3 f''(x) \, dx = \left[ f'(x) \right]_1^3 = f'(3) - f'(1) \]

Substituting the values we found earlier:

\[ f'(3) - f'(1) = 1 - \frac{1}{\sqrt{3}} = 1 - \frac{1}{\sqrt{3}} = \frac{\sqrt{3} - 1}{\sqrt{3}} \]

### Step 5: Combine the results
Now we combine the results from both integrals:

\[ \int_2^3 f'(x) f''(x) \, dx + \int_1^3 f''(x) \, dx = -1 + \left( \frac{\sqrt{3} - 1}{\sqrt{3}} \right) \]

### Step 6: Simplify the final expression
To combine these, we can express \(-1\) as \(-\frac{\sqrt{3}}{\sqrt{3}}\):

\[ -\frac{\sqrt{3}}{\sqrt{3}} + \frac{\sqrt{3} - 1}{\sqrt{3}} = \frac{-\sqrt{3} + \sqrt{3} - 1}{\sqrt{3}} = \frac{-1}{\sqrt{3}} \]

### Final Answer
Thus, the final value of the expression is:

\[ -\frac{1}{\sqrt{3}} \]