Use the Mean Value Theorem to prove the following. Using methods
ther than the Mean Value Theorem will yield no marks. Show all
easoning.
(a) when .

Ask by Hodges Wood. in Canada

Mar 25,2025

Question

Use the Mean Value Theorem to prove the following. Using methods ther than the Mean Value Theorem will yield no marks. Show all easoning. (a) $$ \cos x \leq 2 x+1 $$ when $$ x \geq 0 $$.

UpStudy AI · Accepted Answer

To prove that $$ \cos x \leq 2x + 1 $$ for $$ x \geq 0 $$ using the Mean Value Theorem (MVT), we will follow these steps:

1. **Define the Functions**: Let $$ f(x) = 2x + 1 - \cos x $$. We want to show that $$ f(x) \geq 0 $$ for $$ x \geq 0 $$.

2. **Check the Conditions for MVT**: The function $$ f(x) $$ is continuous on the interval $$[0, a]$$ (for any $$ a > 0 $$) and differentiable on the interval $$(0, a)$$.

3. **Calculate $$ f(0) $$**:
   $$
   f(0) = 2(0) + 1 - \cos(0) = 1 - 1 = 0
   $$

4. **Calculate the Derivative $$ f'(x) $$**:
   $$
   f'(x) = 2 + \sin x
   $$
   Since $$ \sin x \geq 0 $$ for $$ x \geq 0 $$, we have:
   $$
   f'(x) \geq 2
   $$
   This means that $$ f'(x) $$ is always positive for $$ x \geq 0 $$.

5. **Apply the Mean Value Theorem**: According to the MVT, since $$ f(x) $$ is continuous on $$[0, a]$$ and differentiable on $$(0, a)$$, there exists some $$ c \in (0, a) $$ such that:
   $$
   f'(c) = \frac{f(a) - f(0)}{a - 0}
   $$
   Since $$ f(0) = 0 $$, we have:
   $$
   f'(c) = \frac{f(a)}{a}
   $$
   Given that $$ f'(c) \geq 2 $$, we can conclude:
   $$
   \frac{f(a)}{a} \geq 2 \implies f(a) \geq 2a
   $$

6. **Evaluate $$ f(a) $$**: Since $$ f(a) = 2a + 1 - \cos a $$, we have:
   $$
   2a + 1 - \cos a \geq 2a
   $$
   This simplifies to:
   $$
   1 - \cos a \geq 0
   $$
   which is true since $$ \cos a \leq 1 $$ for all $$ a $$.

7. **Conclusion**: Therefore, we have shown that $$ f(x) \geq 0 $$ for $$ x \geq 0 $$, which implies:
   $$
   \cos x \leq 2x + 1 \quad \text{for } x \geq 0.
   $$

Thus, the statement $$ \cos x \leq 2x + 1 $$ is proven for $$ x \geq 0 $$ using the Mean Value Theorem.
To prove that $$ \cos x \leq 2x + 1 $$ for $$ x \geq 0 $$ using the Mean Value Theorem: 1. Define $$ f(x) = 2x + 1 - \cos x $$. 2. Show $$ f(0) = 0 $$. 3. Find $$ f'(x) = 2 + \sin x \geq 2 $$ for $$ x \geq 0 $$. 4. Apply MVT to conclude $$ f(a) \geq 2a $$ for $$ a > 0 $$. 5. Therefore, $$ \cos x \leq 2x + 1 $$ holds for $$ x \geq 0 $$.

Use the Mean Value Theorem to prove the following. Using methods
ther than the Mean Value Theorem will yield no marks. Show all
easoning.
(a) when .

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Use the Mean Value Theorem to prove the following. Using methods ther than the Mean Value Theorem will yield no marks. Show all easoning. (a) when .