Discuss why the equation f(x)=t has no real solutions if t<x0, where x0 is the critical value of f(x)=8x^2-lnx?

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We'll determine first the critical points.

f'(x) = 16x - 1/x

To determine the critical points, we'll put f'(x)=0

16x - 1/x = 0

16x^2 - 1 = 0

Since it is a difference of 2 squares, we'll substitute it by the equivalent product.


Now, we'll set each factor as zero:

4x-1 = 0

x = 1/4

4x+1 = 0

x = -1/4

Since the function f(x) contains the term ln x, we'll impose the constraint that the domain of the function is (0 , +infinite).

Since the value x = -1/4 is not included in the domain (0,+infinite), we'll reject it.

The only critical point is x = 1/4.

We'll calculate the 2nd derivative to see if the extreme is a minimum or maximum point.

f"(x) = 16+ 1/x^2

It is obvious that f"(x)>0, so f(1/4) = 1/2 + ln 4 is a minimum point.

Now, we'll discuss the requested case: f(x)=t has no real solutions if t<x0 = 1/4

Since f(1/4) represents the minimum point of the function, then, it is no  possible to have values of the given function smaller than the minimum point, so the equation f(x)=t has no real solutions.