Let M and N be the number of points on the curve y5 – 9xy + 2x = 0, where the tangents to the curve are parallel to x-axis and y-axis, respectively. Then the value of M + N equals ________.
Correct Answer: 2
Solution and Explanation
The correct answer is 2
Here equation of curve is
y5 – 9xy + 2x = 0 …(i)
On differentiating:
\(5y^4 \frac{dy}{dx} - 9y - 9x \frac{dy}{dx} + 2 = 0\)
∴ \(\frac{dy}{dx} = \frac{9y - 2}{5y^4 - 9x}\)
When tangents are parallel to x axis then 9y – 2 = 0
∴ M = 1.
For tangent perpendicular to x-axis
5y4 – 9x = 0 …(ii)
From equation (1) and equation (2) we get only one point.
∴ N = 1.
∴ M + N = 2.
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Concepts Used:
Application of Derivatives
Various Applications of Derivatives-
Rate of Change of Quantities:
If some other quantity ‘y’ causes some change in a quantity of surely ‘x’, in view of the fact that an equation of the form y = f(x) gets consistently pleased, i.e, ‘y’ is a function of ‘x’ then the rate of change of ‘y’ related to ‘x’ 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) ≤ 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) ≥ 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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