Given
f
(
x
)
=
∫
x
0
t
2
−
25
1
+
cos
2
(
t
)
d
t
 
At what value of  
x
  does the local max of  
f
(
x
)
  occur?

We are given the function:

\[
f(x) = \int_0^x \frac{t^2 - 25}{1 + \cos^2(t)} \, dt
\]

To find where the local maximum of \( f(x) \) occurs, we need to follow these steps:

### Step 1: Compute the derivative of \( f(x) \)
By the Fundamental Theorem of Calculus, the derivative of an integral function is the integrand evaluated at the upper limit:

\[
f'(x) = \frac{x^2 - 25}{1 + \cos^2(x)}
\]

### Step 2: Find critical points
The critical points occur when \( f'(x) = 0 \). So, we set the derivative equal to zero:

\[
\frac{x^2 - 25}{1 + \cos^2(x)} = 0
\]

This equation will be zero when the numerator is zero, because the denominator is always positive:

\[
x^2 - 25 = 0
\]

\[
x^2 = 25
\]

\[
x = \pm 5
\]

### Step 3: Determine which point is a local maximum
To determine whether \( x = 5 \) or \( x = -5 \) corresponds to a local maximum, we examine the second derivative or the sign changes of \( f'(x) \). We compute the second derivative \( f''(x) \):

\[
f''(x) = \frac{d}{dx} \left( \frac{x^2 - 25}{1 + \cos^2(x)} \right)
\]

But instead of calculating \( f''(x) \), we can analyze the sign of \( f'(x) \) around \( x = \pm 5 \):

- For \( x < -5 \), \( x^2 - 25 \) is positive.
- For \( -5 < x < 5 \), \( x^2 - 25 \) is negative.
- For \( x > 5 \), \( x^2 - 25 \) is positive.

Therefore, \( f'(x) \) changes sign from positive to negative at \( x = -5 \), indicating a local maximum at \( x = -5 \). 

So, the local maximum of \( f(x) \) occurs at:

\[
x = -5
\]

Would you like to explore more details on this process or ask any further questions?

Here are 5 questions to deepen understanding:
1. How does the Fundamental Theorem of Calculus apply to integral functions like this?
2. What happens if \( x = 5 \) is evaluated for a local minimum instead?
3. Why is \( f(x) \) continuous and differentiable, and how does this impact finding maxima?
4. Could we use a numerical method to analyze the behavior of the function more closely?
5. How does the second derivative test apply to confirm the nature of critical points?

**Tip**: Always check sign changes in the derivative to confirm maxima or minima when analyzing critical points.

Given f(x) = ∫₀ˣ (t² - 25)/(1 + cos²(t)) dt, at what value of x does the local max of f(x) occur?

Explore how to find the local maximum of the function f(x) = ∫₀ˣ (t² - 25)/(1 + cos²(t)) dt. This solution uses key calculus concepts, such as the Fundamental Theorem of Calculus and critical points analysis. Suitable for students in Grades 11-12 or college-level calculus.

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