# Inequality of cubic and exponential functions

• MHB
• anemone
In summary, a cubic function is a polynomial function with a degree of 3, while an exponential function has a variable in the exponent and a constant base. To graph a cubic function, plot points and connect them with a smooth curve, while an exponential function has a rapidly increasing or decreasing curve. The key characteristics of a cubic function include 3 turning points and either 1 or 3 real roots, with end behavior determined by the leading coefficient. To determine if a function is cubic or exponential, look at the form of the function. Real-world applications of cubic and exponential functions include modeling the volume of a cube, population growth, spread of diseases, decay of radioactive materials, and growth of investments.
anemone
Gold Member
MHB
POTW Director
Prove that $3^n\ge(n+3)^3$ for any natural number $n\ge6$.

This problem is same as
$3^{n-3} > = n^3$ or $3^n >= 27n^3$ for $n >=9$
To prove the same we use principle of mathematical induction

Base step
For n = 9 $LHS = 3^9 = 3^3 * 3^6 = 27 * 9^3$ so $3^n >= 27n^3$
So base step is true
Now $(\frac{n+1}{n})^3$ decreases as n increases and at n = 9 we have $(\frac{n+1}{n})^3= \frac{1000}{729}< 3$
So $(\frac{n+1}{n})^3< 3$ for all $n>=9$
Or $3 > (\frac{k+1}{k})^3\cdots(1)$ for all $k>=9$

Induction step
Let it be true for n = k $k >=9$
We need to prove it to be true for n = k+ 1
$3^k > = 27 k^3$
Multiplying by (1) on both sides
$3^{k+1} > = 27 * (\frac{k+1}{k})^3 * k^3$
Or $3^{k+1} >= 27(k+1)^3$
So it is true for n = k+ 1
We have proved the induction step also

Hence proved

## 1. What is the difference between cubic and exponential functions?

Cubic and exponential functions are both types of mathematical functions, but they differ in the way they grow or decrease. Cubic functions have a polynomial form of ax^3 + bx^2 + cx + d, where the highest power of x is 3. Exponential functions have a form of a^x, where a is a constant and x is the exponent.

## 2. How do the graphs of cubic and exponential functions differ?

The graphs of cubic and exponential functions have distinct shapes. Cubic functions have a characteristic "S" shape, while exponential functions have a curved shape that increases or decreases rapidly. Additionally, the y-intercept of cubic functions is a constant value, while the y-intercept of exponential functions is always 1.

## 3. Which type of function grows faster, cubic or exponential?

Exponential functions grow faster than cubic functions. This is because the exponent in an exponential function increases at a faster rate compared to the power of x in a cubic function. As a result, exponential functions have a steeper slope and grow more rapidly.

## 4. Can cubic and exponential functions intersect?

Yes, cubic and exponential functions can intersect at one or more points. However, the intersection points are limited and depend on the specific values of the constants in each function. In general, exponential functions will grow at a faster rate and eventually overtake cubic functions.

## 5. How are cubic and exponential functions used in real life?

Cubic and exponential functions are used in various fields, including finance, science, and engineering. In finance, exponential functions are used to model compound interest and population growth. Cubic functions are used to model the trajectory of projectiles in physics and to analyze data in statistics. They are also used in computer graphics to create 3D models and animations.

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