To solve the problem described in this exercise:

We are tasked with proving that the equation $f(x) = 0$ has at least seven roots in the interval $[0, 1]$, given the properties of $f$, a continuous function on $[0,1]$:

1. $f(0) = f(1) = 0$,
2. $f(x + \frac{3}{10}) \neq f(x)$ for all $x \in [0, \frac{7}{10}]$.

Solution Outline

Step 1: Analyze the Functional Properties

1. From the second condition $f(x + \frac{3}{10}) \neq f(x)$ for $x \in [0, \frac{7}{10}]$, we can deduce that $f$ is not periodic over this interval and takes distinct values at points spaced $\frac{3}{10}$ apart.

2. The function $f(x)$ must satisfy $f(0) = 0$ and $f(1) = 0$. Because $f$ is continuous, the Intermediate Value Theorem guarantees that if $f$ changes sign in subintervals of $[0,1]$, there will be roots.

Step 2: Divide the Interval $[0,1]$

Divide the interval $[0,1]$ into overlapping subintervals of length $\frac{3}{10}$:

• $[0, \frac{3}{10}]$,
• $[\frac{1}{10}, \frac{4}{10}]$,
• $[\frac{2}{10}, \frac{5}{10}]$,
• $[\frac{3}{10}, \frac{6}{10}]$,
• $[\frac{4}{10}, \frac{7}{10}]$,
• $[\frac{5}{10}, \frac{8}{10}]$,
• $[\frac{6}{10}, 1]$.

Each of these intervals has length $\frac{3}{10}$, and the second condition ensures that $f(x + \frac{3}{10}) \neq f(x)$ within these intervals.

Step 3: Count Roots

• By the Intermediate Value Theorem, since $f(0) = f(1) = 0$, and $f$ is continuous, there must be at least one root in each of these intervals.

• Since there are seven overlapping intervals and $f$ cannot repeat values, $f(x) = 0$ must have at least seven distinct roots in $[0,1]$.

Conclusion:

The equation $f(x) = 0$ has at least seven roots in the interval $[0,1]$.

Do you want further clarification on any step or additional examples?

Related Questions to Expand Understanding:

1. What does the Intermediate Value Theorem state, and how is it applied here?
2. Why does the condition $f(x + \frac{3}{10}) \neq f(x)$ ensure distinct roots?
3. How would the solution change if the interval was $[0,2]$?
4. Could a function with a different continuity condition (e.g., discontinuous) still satisfy this property?
5. How can you generalize this problem for intervals other than $\frac{3}{10}$?

Tip:

Always check for overlapping intervals and use symmetry or periodicity constraints to ensure all roots are distinct.