16. Let $$ h(x) $$ be a function defined for all $$ x
eq 0 $$ such that $$ h(4)=-3 $$ and the derivative of $$ h $$ is given by $$ h^{\prime}(x)=\frac{x^{2}-2}{x} $$ for all $$ x
eq 0 $$. (a) Find equation of the tangent line to the graph $$ y=h(x) $$ at $$ x=4 $$. (2 points) (b) Find all values of $$ x $$ for which the graph of $$ h $$ has a horizontal tangent. (2 points) (c) Find $$ h^{\prime \prime}(x) $$. (2 points)

Question

16. Let $$ h(x) $$ be a function defined for all $$ x 
eq 0 $$ such that $$ h(4)=-3 $$ and the derivative of $$ h $$ is given by $$ h^{\prime}(x)=\frac{x^{2}-2}{x} $$ for all $$ x 
eq 0 $$. (a) Find equation of the tangent line to the graph $$ y=h(x) $$ at $$ x=4 $$. (2 points) (b) Find all values of $$ x $$ for which the graph of $$ h $$ has a horizontal tangent. (2 points) (c) Find $$ h^{\prime \prime}(x) $$. (2 points)

UpStudy AI · Accepted Answer

**Problem 16 (c): Find $$ h''(x) $$.**

To find the second derivative $$ h''(x) $$, we start with the first derivative:
$$ h'(x) = \frac{x^2 - 2}{x} = x - \frac{2}{x} $$

Now, take the derivative of $$ h'(x) $$ with respect to $$ x $$:
$$
h''(x) = \frac{d}{dx} \left( x \right) - \frac{d}{dx} \left( \frac{2}{x} \right) = 1 - \left( -\frac{2}{x^2} \right) = 1 + \frac{2}{x^2}
$$

**Answer:**  
(c) The second derivative is $$ h''(x) = 1 + \dfrac{2}{\,x^{2}\,} $$.
(c) $$ h''(x) = 1 + \frac{2}{x^2} $$.

Let be a function defined for all such that and the derivative of is given by
for all .
(a) Find equation of the tangent line to the graph at . (2 points)
(b) Find all values of for which the graph of has a horizontal tangent. (2 points)
© Find . (2 points)

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(a) Equation of the Tangent Line at

(b) Values of for Horizontal Tangent

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Let be a function defined for all such that and the derivative of is given by for all . (a) Find equation of the tangent line to the graph at . (2 points) (b) Find all values of for which the graph of has a horizontal tangent. (2 points) © Find . (2 points)