Question
(a) Show that if $a$ is a positive constant, then $x=0$ is the only critical point of $f(x)=x+a \sqrt{x}$.(b) Use derivatives to show that $f$ is increasing and its graph is concave down for all $x>0$.
Step 1
Using the power rule for differentiation, we get $f'(x) = 1 + \frac{a}{2\sqrt{x}}$. Show more…
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(a) Show that if $a$ is a positive constant, then $x=0$ is the only critical point of $f(x)=x+a \sqrt{x}$. (b) Use derivatives to show that $f$ is increasing and its graph is concave down for all $x>0$.
(a) If $a$ is a nonzero constant, find all critical points of \[ f(x)=\frac{a}{x^{2}}+x \] (b) Use the second-derivative test to show that if $a$ is positive then the graph has a local minimum, and if $a$ is negative then the graph has a local maximum.
Using the Derivative
Using First and Second Derivatives
(a) If $a$ is a nonzero constant, find all critical points of $$ f(x)=\frac{a}{x^{2}}+x $$ (b) Use the second derivative test to show that if $a$ is positive then the graph has a local minimum, and if $a$ is negative then the graph has a local maximum.
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