Find the relative maximums and minimums of the following functions.
- $$f(x)=x^2+1$$
- $$f(x)=x+5$$
- $$f(x)=\ln(x^2+1)$$
See development and solution
Development:
We will derive the functions and we will make them equal to zero. We will solve the equation and will obtain the solutions (if they exist). Once we know the values where we have maximums and/or minimums (the solutions of the equation) we will use the second derivative to know if they are maximum or minimum.
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We derive the function and equal the result to zero: $$$f'(x)=2x \Rightarrow 2x=0 \Rightarrow x=0$$$ We calculate the second derivative and evaluate it: $$$f''(x)=2 \Rightarrow f''(0)=2>0$$$ Consequently $$x=0$$ is a minimum.
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We derive the function and equal the result to zero: $$$f'(x)=1 \Rightarrow 1=0$$$ Therefore we will not have maximum or minimum.
- We derive the function and equal the result to zero: $$$f'(x)=\dfrac{2x}{x^2+1} \Rightarrow \dfrac{2x}{x^2+1}=0 \Rightarrow 2x=0 \Rightarrow x=0 $$$ We calculate the second derivative and evaluate it: $$$f''(x)=\dfrac{x(x^2+1)-4x^2}{(x^2+1)^2} \Rightarrow f''(0)=\frac{1}{1}=1>0$$$ Consequently, $$x=0$$ is a minimum.
Solution:
- Minimum in $$x = 0$$.
- There are neither maximums nor minimums.
- Minimum in $$x = 0$$.