Consider the curve given by $x ^ { 2 } + 4 y ^ { 2 } = 7 + 3 x y$.
(a) Show that $\frac { d y } { d x } = \frac { 3 y - 2 x } { 8 y - 3 x }$.
(b) Show that there is a point $P$ with $x$-coordinate 3 at which the line tangent to the curve at $P$ is horizontal. Find the $y$-coordinate of $P$.
(c) Find the value of $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ at the point $P$ found in part (b). Does the curve have a local maximum, a local minimum, or neither at the point $P$ ? Justify your answer.

119- From the relation $y^2 + xy^2 + x = 7$, the value of $\dfrac{d^2y}{dx^2}$ at the point $(1,2)$ is which of the following?
(1) $\dfrac{3}{4}$ (2) $\dfrac{4}{5}$ (3) $\dfrac{6}{5}$ (4) $\dfrac{3}{2}$

120- The function $f : \mathbb{R} \to \mathbb{R}$ is twice differentiable. For every real number $x$, the function $g(x) = f(4 - x^2)$ is defined. If $f^{-1}(1) = -5$ and $f^{-1}(1) = -1$, and $f''(1) = -1$, what is the value of $g''(\sqrt{3})$?
(1) $-3$ (2) $-2$ (3) $2$ (4) $3$

7. If $x 2 + y 2 = 1$, then :
(A) yy'" - 2(y ' )2+1=0
(B) $y y ^ { \prime \prime } + \left( y ^ { \prime } \right) 2 + 1 = 0$

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(C) $y y \prime \prime = \left( y ^ { \prime } \right) 2 - 1 = 0$
(D) $y y ^ { \prime \prime } + 2 \left( y ^ { \prime } \right) 2 + 1 = 0$

Consider the functions defined implicitly by the equation $y ^ { 3 } - 3 y + x = 0$ on various intervals in the real line. If $x \in ( - \infty , - 2 ) \cup ( 2 , \infty )$, the equation implicitly defines a unique real valued differentiable function $y = f ( x )$. If $x \in ( - 2,2 )$, the equation implicitly defines a unique real valued differentiable function $y = g ( x )$ satisfying $g ( 0 ) = 0$.
If $f ( - 10 \sqrt { 2 } ) = 2 \sqrt { 2 }$, then $f ^ { \prime \prime } ( - 10 \sqrt { 2 } ) =$
(A) $\frac { 4 \sqrt { 2 } } { 7 ^ { 3 } 3 ^ { 2 } }$
(B) $- \frac { 4 \sqrt { 2 } } { 7 ^ { 3 } 3 ^ { 2 } }$
(C) $\frac { 4 \sqrt { 2 } } { 7 ^ { 3 } 3 }$
(D) $- \frac { 4 \sqrt { 2 } } { 7 ^ { 3 } 3 }$

If $x ^ { 2 } + y ^ { 2 } + \sin y = 4$, then the value of $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ at the point $( - 2,0 )$ is :
(1) $-34$
(2) 4
(3) $-2$
(4) $-32$

If $x ^ { 2 } + y ^ { 2 } + \sin y = 4$, then the value of $\frac { d ^ { 2 } y } { d x ^ { 2 } }$ at the point $( - 2,0 )$ is
(1) - 34
(2) - 32
(3) - 2
(4) 4

If $y ^ { 2 } + \log _ { e } \left( \cos ^ { 2 } x \right) = y , \quad x \in \left( - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right)$ then:
(1) $y \prime \prime ( 0 ) = 0$
(2) $| y \prime ( 0 ) | + | y \prime \prime ( 0 ) | = 1$
(3) $| y \prime \prime ( 0 ) | = 2$
(4) $| y \prime ( 0 ) | + | y \prime \prime ( 0 ) | = 3$

For the curve $C : \left( x ^ { 2 } + y ^ { 2 } - 3 \right) + \left( x ^ { 2 } - y ^ { 2 } - 1 \right) ^ { 5 } = 0$, the value of $3 y ^ { \prime } - y ^ { 3 } y ^ { \prime \prime }$, at the point $( \alpha , \alpha ) , \alpha > 0$, on $C$, is equal to $\_\_\_\_$ .