Real Solutions for 2^x-x^2=1: Finding Roots and Critical Points

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In summary, to solve an exponential equation, you must isolate the variable in the exponent using logarithms. Exponential equations involve a variable in the exponent, while logarithmic equations involve a variable inside the logarithm. An exponential equation can have multiple solutions, which can be checked by substituting the values into the original equation. There are special cases in solving exponential equations, such as when the exponent or base is 0 or 1, or when complex numbers are involved. Special techniques are required to solve these cases.
  • #1
juantheron
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No. of Real solution of ##2^x-x^2 = 1##

Solution (Given as):: Clearly ##x = 0## and ##x = 1## are solution of Given equation.

Formal; define ##f(x) = 2^x - x^2 - 1##, so ##f'(x) = 2^x\ln 2 - 2x##, ##f''(x) = 2^x\ln^2 2 - 2##,

##f'''(x) = 2^x\ln^3 2 > 0##. Study the variation, in order to estimate the values of the roots and

critical points

Now I Did not Under How can we check Nature of ##f(x)## using ##f^{'''}(x)##

Please Help me

Thanked
 
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  • #2
Note that f'''(x) is always positive. This tells you something about f''(x).

Draw a simple graph of these derivatives.
 

FAQ: Real Solutions for 2^x-x^2=1: Finding Roots and Critical Points

How do you solve an exponential equation?

To solve an exponential equation, you must isolate the variable in the exponent by using the inverse operation of exponentiation, which is logarithms. Take the logarithm of both sides of the equation and then use algebraic techniques to solve for the variable.

What is the difference between an exponential equation and a logarithmic equation?

Exponential equations involve a variable in the exponent, while logarithmic equations involve a variable inside the logarithm. The two are related through the inverse operation of exponentiation and logarithms.

Can an exponential equation have more than one solution?

Yes, an exponential equation can have more than one solution. This typically occurs when the base of the exponential function is a fraction or if the equation involves logarithms, which can have multiple solutions.

How do you check if a solution to an exponential equation is correct?

To check if a solution to an exponential equation is correct, simply substitute the value into the original equation and see if it satisfies the equation. If it does, then it is a valid solution.

Are there any special cases in solving exponential equations?

Yes, there are a few special cases when solving exponential equations. This includes cases where the exponent is 0 or 1, where the base is 0 or 1, and when the equation involves complex numbers. These cases require special techniques to solve.

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