1. ## Intrepreting Curve Sketching Conditions

Interpret each piece of information, so that you can be able to sketch the graph of a function $\displaystyle f$ that satisfies the following conditions:

.(a) $\displaystyle f(0) = 0$

.(b) $\displaystyle f''(x) > 0$, . $\displaystyle x \neq 0$

.(c) $\displaystyle \mathop {\lim }\limits_{x \to 0^-} f'(x) = \infty$

.(c) $\displaystyle \mathop {\lim }\limits_{x \to 0^+} f'(x) = -\infty$

.(d) $\displaystyle \mathop {\lim }\limits_{x \to -\infty} f(x) = -\infty$

.(d) $\displaystyle \mathop {\lim }\limits_{x \to \infty} f(x) = \infty$

.

My interpretations are as follows:

.(a) y-intersect at (0,0)

.(b) Concave up for all values of $\displaystyle x$, except at $\displaystyle x = 0$ (possibly an asymptote there?)

.(c) As you approach 0 from the left, the slope approaches infinity and so the function gets higher.

.(c) As you approach 0 from the right, the slope approaches negative infinity and so the function gets lower.

.(d) As you approach negative infinity (left), the function gets lower and lower,

.(d) As you approach positive infinity (right), the function gets higher and higher.

I'm not to sure about them though, and it seems like (b) and (c) are contradictory (if C is true, that would mean concave down to the right of 0)

I would really appreciate a second opinion.

[NOTE: I do not need help with the sketching part, just with interpreting the conditions that $\displaystyle f$ must satisfy.]

2. Originally Posted by Some_One
Sketch the graph of a function $\displaystyle f$ that satisfies the following conditions:

.(a) $\displaystyle f(0) = 0$

.(b) $\displaystyle f''(x) > 0$, . $\displaystyle x \neq 0$

.(c) $\displaystyle \mathop {\lim }\limits_{x \to 0^-} f'(x) = \infty$

.(c) $\displaystyle \mathop {\lim }\limits_{x \to 0^+} f'(x) = -\infty$

.(d) $\displaystyle \mathop {\lim }\limits_{x \to -\infty} f(x) = -\infty$

.(d) $\displaystyle \mathop {\lim }\limits_{x \to \infty} f(x) = \infty$

Interpret each piece of information.

.

My interpretations are as follows:

.(a) y-intersect at (0,0)

.(b) Concave up for all values of $\displaystyle x$, except at $\displaystyle x = 0$ (possibly an asymptote there?)

.(c) As you approach 0 from the left, the slope approaches infinity and so the function gets higher.

.(c) As you approach 0 from the right, the slope approaches negative infinity and so the function gets lower.

.(d) As you approach negative infinity (left), the function gets lower and lower,

.(d) As you approach positive infinity (right), the function gets higher and higher.

I'm not to sure about them though, and it seems like (b) and (c) are contradictory (if C is true, that would mean concave down to the right of 0)

I would really appreciate a second opinion.
If and only if the second condition at d is

$\displaystyle \mathop {\lim }\limits_{x \to \infty} f(x) = -\infty$

then the function could be:

$\displaystyle f(x)=-\sqrt{|x|}$

3. Originally Posted by earboth
If and only if the second condition at d is

$\displaystyle \mathop {\lim }\limits_{x \to \infty} f(x) = -\infty$

then the function could be:

$\displaystyle f(x)=-\sqrt{|x|}$

But it's not. For the assignment we can't change the given conditions. They are what they are - unless you can without a doubt say that they are contradictory, then I can dispute the question with my teacher.

The main part of the assignment is to interpret the meaning of the conditions. Then we just have to roughly sketch the curve based on those four things.

We do not have to find the equation of the function.