\(y=f(x)\) is a quadratic function passing through (β1, 0) and tangent to it at (1, 1) is \(y=x\). Find x intercept by normal at point (π, π + 1), (π > 0)
- 7
- -7
- 5
- -5
The Correct Option is A
Solution and Explanation
Top Questions on Application of derivatives
- Consider the following three statements for the function $f : (0,\infty) \rightarrow \mathbb{R}$ defined by
\[ f(x) = \left| \log_e x \right| - |x - 1| : \]
(I) $f$ is differentiable at all $x>0$.
(II) $f$ is increasing in $(0,1)$.
(III) $f$ is decreasing in $(1,\infty)$.
Then,
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- Let \[ f(x)=x^{2025}-x^{2000},\quad x\in[0,1] \] and the minimum value of the function $f(x)$ in the interval $[0,1]$ be \[ (80)^{80}(n)^{-81}. \] Then $n$ is equal to
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- f(x) = x\(^{2025}\) - x\(^{2000}\), x \(\in\) [0, 1], then minimum value of f(x) is :
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- The slope of the normal to the curve y = \(2x^2\) at x = 1 is:
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- The total cost C(x) in Rupees associated with the production of x units of an item is given by C(x) = \(0.007x^3 - 0.003x^2 + 15x + 400\). The marginal cost when 10 items are produced is:
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Questions Asked in JEE Main exam
In the given figure, the blocks $A$, $B$ and $C$ weigh $4\,\text{kg}$, $6\,\text{kg}$ and $8\,\text{kg}$ respectively. The coefficient of sliding friction between any two surfaces is $0.5$. The force $\vec{F}$ required to slide the block $C$ with constant speed is ___ N.
(Given: $g = 10\,\text{m s}^{-2}$)
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The equivalent resistance between the points \(A\) and \(B\) in the given circuit is \[ \frac{x}{5}\,\Omega. \] Find the value of \(x\).
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Method used for separation of mixture of products (B and C) obtained in the following reaction is:
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- Consider light travelling from a medium A to medium B separated by a plane interface. If the light undergoes total internal reflection during its travel from medium A to B and the speed of light in media A and B are \(2.4 \times 10^8\) m/s and \(2.7 \times 10^8\) m/s, respectively, then the value of critical angle is :
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Concepts Used:
Application of Derivatives
Various Applications of Derivatives-
Rate of Change of Quantities:
If some other quantity βyβ causes some change in a quantity of surely βxβ, in view of the fact that an equation of the form y = f(x) gets consistently pleased, i.e, βyβ is a function of βxβ then the rate of change of βyβ related to βxβ is to be given by
\(\frac{\triangle y}{\triangle x}=\frac{y_2-y_1}{x_2-x_1}\)
This is also known to be as the Average Rate of Change.
Increasing and Decreasing Function:
Consider y = f(x) be a differentiable function (whose derivative exists at all points in the domain) in an interval x = (a,b).
- If for any two points x1 and x2 in the interval x such a manner that x1 < x2, there holds an inequality f(x1) β€ f(x2); then the function f(x) is known as increasing in this interval.
- Likewise, if for any two points x1 and x2 in the interval x such a manner that x1 < x2, there holds an inequality f(x1) β₯ f(x2); then the function f(x) is known as decreasing in this interval.
- The functions are commonly known as strictly increasing or decreasing functions, given the inequalities are strict: f(x1) < f(x2) for strictly increasing and f(x1) > f(x2) for strictly decreasing.
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