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Francesco Giannino » 10.Curve Sketching


Applications of differentiation: concave and convex functions

In mathematics, a concave function is also synonymously called concave downwards, concave down, convex cap or upper convex.

Definition. Formally, a real-valued function f defined on an interval [a,b] is called concave if, for any two points x and y in its domain and any t in [0,1], we have: f(tx+(1-t)y) \geq f(tx)+(1-t)f(y)

This definition merely states that for every z between x and y, the point (z, f(z)) on the graph of f is above the straight line joining the points (x, f(x)) and (y, f(y)).

Applications of differentiation: concave and convex functions


Applications of differentiation: concave and convex functions

Suppose that f is differentiable two times on an interval [a,b].

Theorem. (Concavity Test). If f”’> (x) > 0 for all x on an interval [a,b], then the graph of f is concave upward on [a,b]. If f”> (x) < 0 for all x in [ a,b], then the graph of f is concave downward on [a,b].

Graphing Checklist

To graph a function f, follow these steps:

  1. Find when f is positive, negative, zero, not defined.
  2. Find f’ and form its sign chart. Conclude information about increasing/decreasing and local max/min.
  3. Find f”(if it is possible!) and form its sign chart. Conclude concave up/concave down and inflection.
  4. Put together a big chart to assemble monotonicity and concavity data.
  5. Graph the assigned function.

Exercises

Exercise 1. Take the cubic function f(x)=2x^{3}-3x^{2}-12x and determine its graph.

Solution. (Step 1) The function f(x) is defined for all real numbers. Hence, its domain is: ∀x ∈ ] – ∞, + ∞ [. Now, let’s find the zeros, if it is not complicated. We can at least factor out one power of x:
f(x)=x(2x^{2}-3x-12) so f(0) = 0. The other factor is a quadratic, so the other two roots are x=\frac{3\pm \sqrt{105}}{4}.

We can skip this step for now since the roots are so complicated.

Exercises

(Step 2). Monotonicity.
To apply the first derivative test for extrema, we solve the following inequality:

f'(x)=6x^{2}-6x-12\geq 0 \Rightarrow 6(x+1)(x-2)\geq 0

\Rightarrow x\geq -1 \,\,\text{and}\,\,x\geq 2

and evaluate the sign of f’

\text{The function}\,\,f\,\,\text{is decreasing on}\,\,]-1,2 [\,\,

\text{and is increasing on}\,\,]-\infty,-1[\cup]2,+\infty[


Exercises

Then, applying the first derivative test for extrema, we say:

f(-1)=7 \,\,\text{is a local maximum}
f(2)=-20 \,\,\text{is a local minimum}

Moreover, we study the behavior of the function f at the end points of its domain ]- ∞, + ∞ [:

\lim_{x\rightarrow - \infty}f(x)=\lim_{x\rightarrow - \infty}2x^{3}-3x^{2}-12x=

=\lim_{x\rightarrow - \infty}2x^{3}= -\infty

\text{and}

\lim_{x\rightarrow + \infty}f(x)=\lim_{x\rightarrow + \infty}2x^{3}-3x^{2}-12x=

=\lim_{x\rightarrow + \infty}2x^{3}= +\infty

Showing that the horizontal and oblique asymptotes for the graph of f do not exist.
(Step 4.) Assemble monotonicity data

Exercises

(Step 3). Concavity.
To apply the first concavity test for inflections, we solve the following inequality:

f''(x) = 12x _ 6 = 6(2x _ 1)\geq 0

and evaluate the sign of f”.
(Step 4.) Assemble concavity data

\text{The function}\,\,f\,\,\text{is concave down on}\,\,]-\infty,1/2[\,\,<br />

\text{and is concave up on}\,\,]1/2,+\infty[

f(1/2)=7 \,\,\text{is an inflection point}


Exercises

(Step 5). Graph the function.


Exercises

Exercise 2. Let the quartic function f(x)=2x^{4}-10 and determine its graph.

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