Question

How do you determine the minimum value of the function $f(x)={{4}^{x}}-8$ ?

Answer 1

Hint: In this question, we have to find the minimum value of a function. Thus, we will use the derivative formula to get the solution. First, we will find the first derivative of the given equation using the formula ${{\left( {{a}^{x}} \right)}^{\prime }}={{a}^{x}}\log a$ and ${a}'=0$. After that, we will put the derivative equal to 0, to get the value of x. Then, we will again find the derivative of the first derivative. In the last, we will substitute the value of x in the function, to get the required solution for the problem.

Complete step-by-step solution:
According to the question, we have to find the minimum value of a function.
Thus, we will apply the derivative formula to get the solution.
The function given to us is $f(x)={{4}^{x}}-8$ ---------- (1)
Now, we will find the first derivative of equation (1) with respect to x that is using the derivative of ${{\left( {{a}^{x}} \right)}^{\prime }}={{a}^{x}}\log a$ and ${a}'=0$ , thus we get
$\Rightarrow f'(x)={{4}^{x}}.\log 4-0$ --------- (2)
Therefore, we get
$\Rightarrow f'(x)={{4}^{x}}.\log 4$
Now, we will put the value of first derivative equal to 0 that is let f’(x) = 0 to get the value of x, we get
$\Rightarrow {{4}^{x}}\log 4=0$
Thus, we cannot solve the above equation, which implies we do not have any value of x. therefore, there is no minimum value of the given function.
Therefore, for the function $f(x)={{4}^{x}}-8$ , their does not exist any minimum value

Note: While solving this problem, do mention the formula you are using to avoid confusion and mathematical error. Remember, when we do not find any value of x from the first derivative, it implies there does not exist any maximum nor the minimum value.