To solve this problem using the Intermediate Value Theorem, we need to show that there is a root of the equation $e^x = 4 - 3x$ in the specified interval $(0, 1)$.

Step-by-Step Solution:

1. Reformulate the Equation:

   The given equation is: $e^x = 4 - 3x$

   We define a function $f(x)$ such that: $f(x) = e^x - (4 - 3x) = e^x - 4 + 3x$

   We want to show that there is a root for $f(x) = 0$ in the interval $(0, 1)$.

2. Check Continuity:

   The function $f(x) = e^x - 4 + 3x$ is composed of continuous functions (an exponential function and linear functions), so it is continuous on the interval $[0, 1]$.

3. Calculate $f(0)$ and $f(1)$:

   • For $x = 0$: $f(0) = e^0 - 4 + 3(0) = 1 - 4 = -3$

   • For $x = 1$: $f(1) = e^1 - 4 + 3(1) = e - 1$

   We know that $e \approx 2.718$, so: $f(1) \approx 2.718 - 1 = 1.718$

4. Apply the Intermediate Value Theorem:

   Since $f(x)$ is continuous on the interval $[0, 1]$, and we have: $f(0) = -3 < 0 \quad \text{and} \quad f(1) \approx 1.718 > 0,$ there exists some number $c$ in the interval $(0, 1)$ such that: $f(c) = 0$

Conclusion:

By the Intermediate Value Theorem, there is a root of the equation $e^x = 4 - 3x$ in the interval $(0, 1)$.

Filling in the Blanks:

• $f(0) = -3$
• $f(1) = e - 1$
• Since $f(0) < 0 < f(1)$, there is a number $c$ in $(0, 1)$ such that $f(c) = 0$.

Would you like more details on any of these steps?

Here are 5 related questions that might help expand your understanding:

1. What is the Intermediate Value Theorem, and why is it important in mathematics?
2. How do we determine if a function is continuous on a given interval?
3. What are some examples of equations where the Intermediate Value Theorem can be applied?
4. How can we approximate the root found using the Intermediate Value Theorem?
5. What are other methods to find the root of an equation besides the Intermediate Value Theorem?

Tip: Remember, the Intermediate Value Theorem only tells us that a root exists, not the exact value of the root. To find the precise root, numerical methods like the Newton-Raphson method can be used.