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Chapter 4: Problem 8

Find the points on the curve \(x^{2}+x y+y^{2}=7\) at which (i) the tangent is parallel to the \(x\) -axis, (ii) the tangent is parallel to the \(y\) -axis.

Short Answer

Expert verified
The points on the curve \(x^{2}+xy+y^{2}=7\) with tangent parallel to the x-axis are \((-2\sqrt{\frac{7}{3}},\sqrt{\frac{7}{3}})\) and \((2\sqrt{\frac{7}{3}},-\sqrt{\frac{7}{3}})\). The points on the curve with tangent parallel to the y-axis are \((\sqrt{\frac{7}{3}}, -2\sqrt{\frac{7}{3}})\) and \((- \sqrt{\frac{7}{3}}, 2\sqrt{\frac{7}{3}})\).

Step by step solution

01

Calculate the first partial derivatives of the equation

Using the implicit function theorem, The equation of the curve is: \(F(x,y) = x^{2}+xy+y^{2}-7 = 0\) Now we find the first partial derivatives of F(x, y) with respect to x and y. \(\frac{\partial F}{\partial x} = 2x + y\) \(\frac{\partial F}{\partial y} = x + 2y\) These two partial derivatives are equivalent to the gradient of the curve.
02

Determine the conditions for the tangent properties

For tangent parallel to the x-axis, the slope will be 0 in the y-direction, i.e., \(\frac{\partial F}{\partial y} = 0\). For tangent parallel to the y-axis, the slope will be 0 in the x-direction, i.e., \(\frac{\partial F}{\partial x} = 0\).
03

Solve for the points with the desired tangent properties

(i) Tangent parallel to the x-axis: We have \(\frac{\partial F}{\partial y} = x + 2y = 0\) Given the original curve equation: \(x^{2} + xy + y^{2} = 7\) By solving the two equations simultaneously: \[x = -2y\] Substitute \(x\) in the original equation: \[(-2y)^{2} - 2y^{2} + y^{2} = 7\] \[3y^{2} = 7\] \[y = \pm \sqrt{\frac{7}{3}}\] Substitute the y values in the x value equation: When \(y=\sqrt{\frac{7}{3}}\), \(x=-2\sqrt{\frac{7}{3}}\). When \(y=-\sqrt{\frac{7}{3}}\), \(x=2\sqrt{\frac{7}{3}}\). So, the points are \((-2\sqrt{\frac{7}{3}},\sqrt{\frac{7}{3}})\) and \((2\sqrt{\frac{7}{3}},-\sqrt{\frac{7}{3}})\). (ii) Tangent parallel to the y-axis: We have \(\frac{\partial F}{\partial x} = 2x + y = 0\) Given the original curve equation: \(x^{2} + xy + y^{2} = 7\) By solving the two equations simultaneously: \[y = -2x\] Substitute \(y\) in the original equation: \[x^{2} - 2x^{2} + 4x^{2} = 7\] \[3x^{2} = 7\] \[x = \pm \sqrt{\frac{7}{3}}\] Substitute the x values in the y value equation: When \(x=\sqrt{\frac{7}{3}}\), \(y=-2\sqrt{\frac{7}{3}}\). When \(x=-\sqrt{\frac{7}{3}}\), \(y=2\sqrt{\frac{7}{3}}\). So, the points are \((\sqrt{\frac{7}{3}}, -2\sqrt{\frac{7}{3}})\) and \((- \sqrt{\frac{7}{3}}, 2\sqrt{\frac{7}{3}})\). In conclusion, the points on the curve with tangent parallel to the x-axis are \((-2\sqrt{\frac{7}{3}},\sqrt{\frac{7}{3}})\) and \((2\sqrt{\frac{7}{3}},-\sqrt{\frac{7}{3}})\). And the points on the curve with tangent parallel to the y-axis are \((\sqrt{\frac{7}{3}}, -2\sqrt{\frac{7}{3}})\) and \((- \sqrt{\frac{7}{3}}, 2\sqrt{\frac{7}{3}})\).

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Most popular questions from this chapter

(i) Let \(f, g:[a, b] \rightarrow \mathbb{R}\) be continuous on \([a, b]\) and differentiable on \((a, b)\). If \(f(a) \leq g(a)\) and \(f^{\prime}(x) \leq g^{\prime}(x)\) for all \(x \in(a, b)\), then show that \(f(b) \leq g(b)\) (ii) Use (i) to show that \(15 x^{2} \leq 8 x^{3}+12 \leq 18 x^{2}\) for all \(x \in[1.25,1.5]\). Deduce that the range of the function \(h:[1.25,1.5] \rightarrow \mathbb{R}\) given by \(h(x)=\left(2 x^{3}+3\right) / 3 x^{2}\) is contained in \([1.25,1.5]\).

Let \(P_{1}=\left(x_{1}, y_{1}\right)\) and \(P_{2}=\left(x_{2}, y_{2}\right)\) be two points on the curve \(y=\) \(a x^{2}+b x+c .\) If \(P_{3}=\left(x_{3}, y_{3}\right)\) lies on the arc \(P_{1} P_{2}\) and the tangent to the curve at \(P_{3}\) is parallel to the chord \(P_{1} P_{2}\), show that \(x_{3}=\left(x_{1}+x_{2}\right) / 2\).

Let \(m \in \mathbb{N}\) and \(f, g:[a, b] \rightarrow \mathbb{R}\) be such that \(f, f^{\prime}, \ldots, f^{(m-1)}\) as well as \(g, g^{\prime}, \ldots, g^{(m-1)}\) are continuous on \([a, b]\) and \(f^{(m)}, g^{(m)}\) exist on \((a, b)\). Suppose \(f^{\prime}(a)=f^{\prime \prime}(a)=\cdots=f^{(m-1)}(a)=0\) and \(g^{\prime}(a)=g^{\prime \prime}(a)=\cdots=\) \(g^{(m-1)}(a)=0\), but \(g^{(m)}(x) \neq 0\) for all \(x \in(a, b)\). Prove that there exist \(c_{1}, \ldots, c_{m} \in(a, b)\) such that $$ \frac{f(b)-f(a)}{g(b)-g(a)}=\frac{f^{\prime}\left(c_{1}\right)}{g^{\prime}\left(c_{1}\right)}=\frac{f^{\prime \prime}\left(c_{2}\right)}{g^{\prime \prime}\left(c_{2}\right)}=\cdots=\frac{f^{(m)}\left(c_{m}\right)}{g^{(m)}\left(c_{m}\right)} $$

Let a function \(f:[a, b] \rightarrow \mathbb{R}\) be continuous and its second derivative \(f^{\prime \prime}\) exist everywhere on the open interval \((a, b) .\) Suppose the line segment joining \((a, f(a))\) and \((b, f(b))\) intersects the graph of \(f\) at a third point (c, \(f(c))\), where \(a

Let \(I\) be an interval containing more than one point and \(f: I \rightarrow \mathbb{R}\) be any function. (i) Assume that \(f\) is differentiable. If \(f^{\prime}\) is nonnegative on \(I\) and \(f^{\prime}\) vanishes at only a finite number of points on any bounded subinterval of \(I\), then show that \(f\) is strictly increasing on \(I\). (ii) Assume that \(f\) is twice differentiable. If \(f^{\prime \prime}\) is nonnegative on \(I\) and \(f^{\prime \prime}\) vanishes at only a finite number of points on any bounded subinterval of \(I\), then show that \(f\) is strictly convex on \(I\). (iii) Consider \(f: \mathbb{R} \rightarrow \mathbb{R}\) given by \(f(x)=(x-2 n)^{3}+2 n\), where \(n \in \mathbb{Z}\) is such that \(x \in[2 n-1,2 n+1) .\) Show that \(f\) is differentiable on \(\mathbb{R}\) and \(f^{\prime \prime}\) exists on \((2 n-1,2 n+1)\), but \(f_{+}^{\prime \prime}(2 n+1)=6\), whereas \(f_{-}^{\prime \prime}(2 n+1)=-6\) for each \(n \in \mathbb{N} .\) Also show that \(f\) is strictly increasing on \(\mathbb{R}\) although \(f^{\prime}(2 n)=0\) for each \(n \in \mathbb{N}\). (Compare (i) above and Exercise 12 in the list of Revision Exercises at the end of Chapter 7.) (iv) Consider \(g: \mathbb{R} \rightarrow \mathbb{R}\) given by \(g(x)=(x-2 n)^{4}+8 n x\), where \(n \in \mathbb{Z}\) is such that \(x \in[2 n-1,2 n+1) .\) Show that \(g\) is twice differentiable on \(\mathbb{R}\) and \(g^{\prime \prime \prime}\) exists on \((2 n-1,2 n+1)\), but \(g_{+}^{\prime \prime \prime}(2 n+1)=24\), whereas \(g_{-}^{\prime \prime \prime}(2 n+1)=-24\) for each \(n \in \mathbb{N} .\) Also show that \(g\) is strictly convex on \(\mathbb{R}\) although \(g^{\prime \prime}(2 n)=0\) for each \(n \in \mathbb{N}\). (Compare (ii) above and Exercise 13 in the list of Revision Exercises at the end of Chapter \(7 .\) )
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