4.1: Maxima and Minima
- Page ID
- 144280
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- A continuous function \(f(x)\) is defined over an interval \([-2, 3]\). Can we guarantee that \(f(x)\) has an absolute maximum over \([-2, 3]\)? If yes, briefly explain. If not, sketch a counterexample.
- A function \(g(x)\) is defined over an interval \([-2, 3]\). Can we guarantee that \(g(x)\) has an absolute maximum over \([-2, 3]\)? If yes, briefly explain. If not, sketch a counterexample.
- A discontinuous function \(h(x)\) is defined over an interval \([-2, 3]\). Can we guarantee that \(h(x)\) does not have an absolute maximum over \([-2, 3]\)? If yes, briefly explain. If not, sketch a counterexample.
- A continuous function \(f(x)\) is defined over an interval \([-2, \infty)\). Can we guarantee that \(f(x)\) has an absolute maximum over \([-2, \infty)\)? If yes, briefly explain. If not, sketch a counterexample.
- A discontinuous function \(h(x)\) is defined over an interval \([-2, \infty)\). Can we guarantee that \(h(x)\) does not have an absolute maximum over \([-2, \infty)\)? If yes, briefly explain. If not, sketch a counterexample.
- Find the absolute maximum and minimum of \(f(x) = x^2 + 3\) over the interval \([-1, 4]\).
- Find the absolute maximum and minimum of \(g(x) = 3 + 2x - x^2\) over the interval \([-1, 1]\).
- Find the absolute maximum and minimum of \(h(x) = x^2 + \dfrac{2}{x}\) over the interval \([1, 4]\).
- Find the absolute maximum and minimum of \(y = \sqrt{x} - \sqrt{x}^3\) over the interval \([0, 4]\).
- Find the absolute maximum and minimum of \(f(x) = \sin x \cos x\) over the interval \([0, 2\pi]\).
- Find the absolute maximum and minimum of \(g(x) = \sin x + \cos x\) over the interval \([0, 2\pi]\).
- Find all local maxima and minima of \(f(x) = x^2 - x\). Confirm your results with a graphing calculator.
- Find all local maxima and minima of \(g(x) = 2 + 3x - x^3\). Confirm your results with a graphing calculator.
- Find all local maxima and minima of \(h(x) = x^4 - 2x^2 + 3\). Confirm your results with a graphing calculator.
- Find all local maxima and minima of \(y = x^4 + 2x^3\). Confirm your results with a graphing calculator.
- Find all local maxima and minima of \(f(x) = \dfrac{x^2-1}{x}\). Confirm your results with a graphing calculator.
- Find all local maxima and minima of \(g(x) = \dfrac{1}{x^2 + 1}\). Confirm your results with a graphing calculator.
- Find all local maxima and minima of \(h(x) = \dfrac{x^2 + x + 6}{x + 1}\). Confirm your results with a graphing calculator.
- Find all local maxima and minima of \(y = \dfrac{1}{x^2 + 1}\). Confirm your results with a graphing calculator.
- Find all local maxima and minima of \(f(x) = \sqrt{4 - x^2}\). Confirm your results with a graphing calculator.
- Find all local maxima and minima of \(g(x) = x^{1/3}\). Confirm your results with a graphing calculator.
- Find all local maxima and minima of \(h(x) = x^{2/3}\). Confirm your results with a graphing calculator.
- Find all local maxima and minima of \(y(x) = |x|\). Confirm your results with a graphing calculator.
- Find all local maxima and minima of \(g(x) = xe^x\). Confirm your results with a graphing calculator.
- Find all local maxima and minima of \(h(x) = x^2e^x\). Confirm your results with a graphing calculator.
- Find all local maxima and minima of \(f(x) = x\ln x\). Confirm your results with a graphing calculator.
- Find all local maxima and minima of \(h(x) = |x-1| + |x+1|\). Confirm your results with a graphing calculator.
- Find all local maxima and minima of \(f(x) = \left\{\begin{align*}
& x^2 + 1, & x \le 1 \\
& x^2 - 4x + 5, & x > 1.
\end{align*}\right.\) Confirm your results with a graphing calculator.