3. Q3b
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Similar Question 1
<p>If <code class='latex inline'>f(1) = 10</code> and <code class='latex inline'>f'(x) \geq 2</code> for <code class='latex inline'>1 \leq x\leq 4</code>, how small can <code class='latex inline'>f(4)</code> possibly be?</p>
Similar Question 2
<p>Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers <code class='latex inline'>c</code> that satisfy the conclusion of the Mean Value Theorem</p><p><code class='latex inline'> \displaystyle f(x) = x^3 -3x + 2, [-2, 2] </code></p>
Similar Question 3
<p>The graph of a function <code class='latex inline'>f</code> is shown.</p><img src="/qimages/2309" /> <ul> <li>Estimate the value(s) of <code class='latex inline'>c</code> that satisfy the conclusion of the Mean Value Theorem on the interval [2, 6].</li> </ul>
Similar Questions
Learning Path
L1 Quick Intro to Factoring Trinomial with Leading a
L2 Introduction to Factoring ax^2+bx+c
L3 Factoring ax^2+bx+c, ex1
Now You Try
<p>Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers <code class='latex inline'>c</code> that satisfy the conclusion of the Mean Value Theorem</p><p><code class='latex inline'> \displaystyle f(x) = x^3 -3x + 2, [-2, 2] </code></p>
<p> Verify that the function satisfies the the hypotheses of Rolle&#39;s Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle&#39;s Theorem.</p><p><code class='latex inline'>f(x) = x \sqrt{x + 6} , [-6, 0]</code></p>
<p>Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers <code class='latex inline'>c</code> that satisfy the conclusion of Rolle’s Theorem.</p><p><code class='latex inline'> \displaystyle f(x) = 2x^2 -4x + 5, [-1, 3] </code></p>
<p>Show that the equation <code class='latex inline'> \displaystyle x^3 -15x + c = 0 </code> has at most one root in the interval [-2, 2].</p>
<p>Suppose that <code class='latex inline'>f</code> is differentiable on R and has two roots. Show that <code class='latex inline'>f'</code> has at least one root.</p>
<p>Let <code class='latex inline'>f(x) = \tan x</code>. Show that <code class='latex inline'>f(0) = f(\pi) </code> but there is no number <code class='latex inline'>c</code> in <code class='latex inline'>(0, \pi)</code>, such that <code class='latex inline'>f'(c) = 0</code>. Why does this not contradict Rolle’s Theorem?</p>
<p>Does there exist a function <code class='latex inline'>f</code> such that <code class='latex inline'>f(0) = -1, f(2) = 4</code>, and <code class='latex inline'>f'(x) \leq 2</code> for all <code class='latex inline'>x</code>?</p>
<p>The graph of a function <code class='latex inline'>f</code> is shown.</p><img src="/qimages/2309" /> <ul> <li>Estimate the value(s) of <code class='latex inline'>c</code> that satisfy the conclusion of the Mean Value Theorem on the interval [2, 6].</li> </ul>
<p>Let <code class='latex inline'>\displaystyle f(x) = \frac{x + 1}{x - 1}</code>. Show that there is no value of <code class='latex inline'>c</code> such that <code class='latex inline'>f(2) - f(0) = f'(c)(2 - 0 )</code>. Why does this not contradict the Mean Value Theorem?</p>
<p>Show that the equation has exactly one real root.</p><p><code class='latex inline'> \displaystyle 2x + \cos x = 0 </code></p>
<p>Suppose f is an odd function and is differentiable every- where. Prove that for every positive number <code class='latex inline'>b</code>, there exists a number <code class='latex inline'>c</code> in <code class='latex inline'>(-b, b)</code> such that <code class='latex inline'>f'(c) = f(b)/b</code></p>
<p> Verify that the function satisfies the the hypotheses of Rolle&#39;s Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle&#39;s Theorem.</p><p><code class='latex inline'>f(x) = \sin 2\pi x, [- 1, 1]</code></p>
<p>Find the number <code class='latex inline'>c</code> that satisfies the conclusion of the Mean Value Theorem on the given interval. Graph the function, the secant line through the endpoints, and the tangent line at <code class='latex inline'>(c, f(c))</code>. Are the secant line and the tangent line parallel?</p><p><code class='latex inline'> \displaystyle f(x) = e^{-x}, [0, 2] </code></p>
<p>Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers <code class='latex inline'>c</code> that satisfy the conclusion of Rolle’s Theorem.</p><p><code class='latex inline'> \displaystyle f(x) =x + 1/x, [\frac{1}{2}, 2] </code></p>
<p>Let <code class='latex inline'>f(x) = 2 - |2x - 1|</code>. Show that there is no value of <code class='latex inline'>c</code> such that <code class='latex inline'>f(3) - f(0) = f'(c)(3 - 9)</code> Why does this not contradict the Mean Value Theorem?</p>
<p>Suppose that <code class='latex inline'>f</code> and <code class='latex inline'>g</code> are continuous on [a, b] and differentiable on <code class='latex inline'>(a, b)</code>. Suppose also that <code class='latex inline'>f(a) = g(a)</code> and <code class='latex inline'>f'(x) < g'(x)</code> for <code class='latex inline'>a < x < b</code>. Prove that <code class='latex inline'>f(b) < g(b)</code>. [Hint: Apply MVT to the function <code class='latex inline'>h = f -g</code>]</p>
<p>Show that the equation <code class='latex inline'>x^3 - 15x + c = 0</code> has at most one root in <code class='latex inline'>[-2, 2]</code>.</p>
<p>Show that<br> <code class='latex inline'> \displaystyle\sin x < x </code> if <code class='latex inline'>0 < x < 2\pi</code></p>
<p>Suppose that <code class='latex inline'>3 \leq f'(x) \leq 5</code> for all values of <code class='latex inline'>x</code>. Show that <code class='latex inline'>18 \leq f(8) -f(2) \leq 30</code>.</p>
<p>Show that the equation has exactly one real root.</p><p><code class='latex inline'> \displaystyle x^3 + e^x = 0 </code></p>
<p>Prove the identity <code class='latex inline'> \displaystyle \arcsin \frac{x - 2}{x + 1} = 2\arctan \sqrt{x} -\frac{\pi}{2} </code>.</p>
<p>Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers <code class='latex inline'>c</code> that satisfy the conclusion of the Mean Value Theorem</p><p><code class='latex inline'> \displaystyle f(x) = \ln x, [1, 4] </code></p>
<p>Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers <code class='latex inline'>c</code> that satisfy the conclusion of the Mean Value Theorem</p><p><code class='latex inline'> \displaystyle f(x) = 1/x, [1, 3] </code></p>
<p>Let <code class='latex inline'>f(x) = (x - 3)^{-2}</code>. Show that there is no value of <code class='latex inline'>c</code> in (1, 4) such that <code class='latex inline'>f(4) -f(1) = f'(c)(4 -1)</code>. Why does this not contradict the Mean Value Theorem?</p>
<p>Show that the equation <code class='latex inline'>ax^5 + bx^3 + c = 0</code>, where <code class='latex inline'>a, b</code>, and <code class='latex inline'>c</code> are constants such that <code class='latex inline'>a,b > 0</code>, has exactly one(real) solution.</p>
<p>Does there exist a function <code class='latex inline'>f</code> such that <code class='latex inline'>f(0) = -1, f(2) = 4</code>, and <code class='latex inline'>f'(x) \leq 2</code> for all <code class='latex inline'>x</code>?</p>
<p>Let <code class='latex inline'>f(x) = |x - 1|</code>. Show that there is no value of <code class='latex inline'>c</code> such that <code class='latex inline'>f(3) - f(0) = f'(c)(3 - 0)</code>. Why does this not contradict the Mean value Theorem?</p>
<p>Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers <code class='latex inline'>c</code> that satisfy the conclusion of Rolle’s Theorem.</p><p><code class='latex inline'> \displaystyle f(x) = x^3 -2x^2 - 4x+ 2, [-2, 2] </code></p>
<p>Verify that the function satisfies the three hypotheses of Rolle’s Theorem on the given interval. Then find all numbers <code class='latex inline'>c</code> that satisfy the conclusion of Rolle’s Theorem.</p><p><code class='latex inline'> \displaystyle f(x) = \sin(x/2), [\pi/2, 3\pi/2] </code></p>
<p>The graph of a function <code class='latex inline'>f</code> is shown.</p><img src="/qimages/2309" /> <ul> <li>Estimate the value(s) of <code class='latex inline'>c</code> that satisfy the conclusion of the Mean Value Theorem on the interval [0, 8].</li> </ul>
<p>Draw the graph of a function defined on [0, 8] such that <code class='latex inline'>f(0) = f(8) = 3</code> and the function does not satisfy the conclusion of Rolle’s Theorem on [0, 8].</p>
<p>Draw the graph of a function that is continuous on [0, 8] where <code class='latex inline'>f(0) = 1</code> and <code class='latex inline'>f(8) = 4</code> and that does not satisfy the conclusion of the Mean Value Theorem on [0, 8].</p>
<p>The graph of a function <code class='latex inline'>f</code> is shown.</p><img src="/qimages/2309" /> <ul> <li>Verify that g satisfies the hypotheses of the Mean Value Theorem on the interval [0, 8].</li> </ul>
<p>Show that the equation <code class='latex inline'>2x - 3 - \sin x = 0</code> has exactly one root.</p>
<p>Find the number <code class='latex inline'>c</code> that satisfies the conclusion of the Mean Value Theorem on the given interval. Graph the function, the secant line through the endpoints, and the tangent line at <code class='latex inline'>(c, f(c))</code>. Are the secant line and the tangent line parallel?</p><p><code class='latex inline'> \displaystyle f(x) = \sqrt{x}, [0, 4] </code></p>
<p>Let <code class='latex inline'>f(x) = 1 -x^{\frac{2}{3}}</code>. Show that <code class='latex inline'>f(-1) = f(1)</code> but there is no number <code class='latex inline'>c</code> in (-1, 1) such that <code class='latex inline'>f'(c) = 0</code>. Why does this not contradict Rolle’s Theorem?</p>
<p>Let <code class='latex inline'>\displaystyle f(x) = 1 - x^{\frac{2}{3}}</code>. Show that <code class='latex inline'>f(-1) = f(1)</code> but there is no number <code class='latex inline'>c</code> in (-1, 1) such that <code class='latex inline'>f'(c) = 0</code>. Why does this not contradict Rolle&#39;s Theorem?</p>
<p>If <code class='latex inline'>f(1) = 10</code> and <code class='latex inline'>f'(x) \geq 3</code> for <code class='latex inline'>1 \leq x \leq 4</code>, how small can <code class='latex inline'>f(4)</code> possibly be?</p>
<p>Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. Then find all numbers <code class='latex inline'>c</code> that satisfy the conclusion of the Mean Value Theorem</p><p><code class='latex inline'> \displaystyle f(x) = 2x^2 -3x + 1, [0, 2] </code></p>
<p>If <code class='latex inline'>f(1) = 10</code> and <code class='latex inline'>f'(x) \geq 2</code> for <code class='latex inline'>1 \leq x\leq 4</code>, how small can <code class='latex inline'>f(4)</code> possibly be?</p>
<p>Show that the equation <code class='latex inline'> \displaystyle x^4 + 4x + c = 0 </code> has at most two roots.</p>
<p> Verify that the function satisfies the the hypotheses of Rolle&#39;s Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle&#39;s Theorem.</p><p><code class='latex inline'>f(x) = x^3 - 3x^2 + 2x + 5, [0, 2]</code></p>
<p>The graph of a function f is shown. Verify that f satis es the hypotheses of Rolle’s Theorem on the interval f0, 8g. Then estimate the value(s) of c that satisfy the conclusion of Rolle’s Theorem on that interval.</p><img src="/qimages/2308" />
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