Sample QuestionsMean Value Theorems questions
One sample from each question group in this chapter. Select any group above to see the full set with answer keys.
If $4a + 2b + c = 0$, then the equation $3ax^2+ 2bx + c = 0$ has atleast one real root lying in the interval:
- A
$(0, 1)$
- B
$(1, 2)$
- ✓
$(0, 2)$
- D
Answer: C.
View full solution →For the function $\text{f}(\text{x})=\text{x}+1\text{x},\text{x}\in[1,3],$ the value of $c$ for the Lagrange's mean value theorem is:
Answer: B.
View full solution →The value of $c$ in Rolle's theorem for the function $\text{f}(\text{x})=\frac{\text{x}(\text{x}+1)}{\text{e}^{\text{x}}}$ defined on $[-1, 0]$ is:
- A
$0.5$
- B
$\frac{1+\sqrt5}{2}$
- ✓
$\frac{1-\sqrt5}{2}$
- D
$-0.5$
Answer: C.
View full solution →When the tangent to the curve $\text{y}=\text{x}\log\text{x}$ is parallel to the chord joining the points $(1, 0)$ and $(e, e),$ the value of $x$ is:
- ✓
$\text{e}^{\frac{1}{1}-\text{e}}$
- B
$\text{e}^{(\text{e}-1)(2\text{e}-1)}$
- C
$\text{e}^{\frac{2\text{e}-1}{\text{e}-1}}$
- D
$\frac{\text{e}-1}{\text{e}}$
Answer: A.
View full solution →The value of $c$ in Rolle's theorem when $f(x) = 2x^3 - 5x^2 - 4x + 3$, $\text{x}\in\Big[\frac{1}{3},3\Big]$ is$:$
- ✓
$2$
- B
$-\frac{1}{3}$
- C
$-2$
- D
$\frac{2}{3}$
Answer: A.
View full solution →Discuss the applicability of the Rolle's theorem for the following function on the indicated interval
$\text{f}(\text{x})=\sin\frac{1}{\text{x}}\text{ for}-1\leq\text{x}\leq1$
View full solution →Discuss the applicability of the Rolle's theorem for the following function on the indicated interval
$\text{f}(\text{x})=2\text{x}^2-5\text{x}+3\text{ on }[1,3]$
View full solution →Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem.
$f(x) = 2x^2 - 3x + 1$ on $[1, 3]$
View full solution →Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem.
$f(x) = x^2 - 3x + 2$ on $[-1, 2]$
View full solution →Discuss the applicability of the Rolle's theorem for the following function on the indicated interval
$\text{f}(\text{x})=3+(\text{x}-2)^{\frac{2}{3}}\text{ on }[1,3]$
View full solution →Verify Lagrange's mean value theorem for the following function on the indicated intervals. find a point 'c' in the indicated interval as stated by the Lagrange's mean value theorem.
$\text{f}(\text{x})=\sqrt{\text{x}^2-4}\text{ on }[2,4]$
View full solution →Show that the Lagrange's mean value theorem is not applicable to the function
$\text{f}(\text{x})=\frac{1}{\text{x}}\text{ on }[-1,1]$
View full solution →Find a point on the curve $y = x^3 + 1$ where the tangent is parallel to the chord joining (1, 2) and (3, 28).
View full solution →Find a point on the curve $y = x^2 + x,$ where the tangent is parallel to the chord joining $(0, 0)$ and $(1, 2)$.
View full solution →Verify Rolle's theorem for the following function on the indicated intervals $f(x) = (x - 1) (x - 2)^2$ on $[1, 2]$
View full solution →Find a point on the parabola $y = (x − 3)^2$, where the tangent is parallel to the chord joining $(3, 0)$ and $(4, 1)$.
View full solution →Verify the hypothesis and conclusion of Lagrange's mean value theorem for the function
$\text{f}(\text{x})=\frac{1}{4\text{x}-1},1\leq\text{x}\leq4.$
View full solution →