Collection of Maths Problems

Shape of a function (with convexity)

Task number: 3049

• Variant 1

  $\displaystyle f(x)=x^3 - 12x + 16$

• Resolution

  Let $f(x) = x^3 - 12x + 16$.

  Domain of definition, range and intersections with axes: The domain of definition of $f$ is all real numbers, and we will determine the function's range later. We can factor the function as $f(x) = (x-2)^2(x+4)$, so it intersects the $x$-axis at the points $(-4{,}0)$ and $(2{,}0)$, and intersects the $y$-axis at the points $(0, 16)$.

  Monotonicity: We have $f'(x) = 3x^2 - 12 = 3(x-2)(x+2)$. So we have critical points at $x = \pm 2$. From the factorization we can easily tell that $f'(x) > 0$ on the intervals $(-\infty, -2)$ and $(2, \infty)$, so the function is increasing on these intervals. Conversely $f'(x) < 0$ on the interval $(-2, 2)$, where the function is decreasing. So we know that at $x = -2$ there is a local maximum, namely $f(-2) = 32$, and at $x = 2$ there is a local minimum, namely $f(2) = 0$.

  Convexity: Now we compute $f''(x) = 6x$. So we have $f''(x) > 0$ for $x \in (0, \infty)$ and $f''(x) < 0$ for $x \in (-\infty, 0)$. So the function is concave on the interval $(-\infty, 0)$ and convex on $(0, \infty)$. At $x = 0$ we have an inflection point.

  Limits at boundary points: We have $\lim_{x \to \infty} f(x)= \infty$ and $\lim_{x \to -\infty} f(x)= -\infty$ (as you can surely easily compute). This helps us sketch the graph around "infinities". And now we also know, since $f$ is continuous, that the range of $f$ is $\mathbb R$.

  Asymptotes: We have $\lim_{x \to \infty} f(x)/x = \lim_{x \to -\infty} f(x)/x =\infty$, as you can surely easily compute. From here it follows that the function does not have asymptotes either at $\infty$ or at $-\infty$.

  

• Result

  Here is the graph, though the other steps presented in the solution (and omitted here) are important.

  

• Variant 2

  $\displaystyle f(x)=\arcsin(\cos x)$

• Variant 3

  $\displaystyle f(x)=\ln^3(x)$

• Variant 4

  $\displaystyle f(x)=e^{-x^2}$

• Variant 5

  $\displaystyle f(x)=\ln(4{\cdot}3^x + 2)$

• Variant 6

  $\displaystyle f(x)=\ln\left(\left|\tan\frac{x}4\right|\right)$

• Variant 7

  $\displaystyle f(x)=\arcsin\left(\frac{2x}{x^2 + 1}\right)$

• Variant 8

  $\displaystyle f(x)=x^2 e^{-x}$

• Variant 9

  $\displaystyle f(x)=\frac{x^3}{(x-2)^2}$

• Variant 10

  $\displaystyle f(x)=x-\ln(x+1)$

• Variant 11

  $\displaystyle f(x)=x\, e^{-|x-1|}$