Find the exact location of all the relative arid absolute extrema of the function. (Orc \[ g(x)=5 x^{2}-20 \sqrt{x} \] $g$ has - Select--- at $(x, y)=$ $g$ has --Select-- at $(x, y)=$ Submit Answer
Solution
Step 1 ：First, we find the derivative of the function \(g(x)=5 x^{2}-20 \sqrt{x}\). The derivative is \(g'(x) = 10x - \frac{10}{\sqrt{x}}\).
Step 2 ：Next, we set the derivative equal to zero and solve for x to find the critical points. The critical point is \(x = 1\).
Step 3 ：We substitute \(x = 1\) into the original function to get the corresponding y-value. The y-value is \(y = -15\). Therefore, the function has an extrema at \((1, -15)\).
Step 4 ：To determine whether this is a relative or absolute extrema, and whether it is a maximum or minimum, we use the second derivative test. The second derivative of the function is \(g''(x) = 10 + \frac{5}{x^{\frac{3}{2}}}\).
Step 5 ：The second derivative at the critical point is positive, which means the function is concave up at this point. Therefore, the function has a relative minimum at \((1, -15)\).
Step 6 ：Since this is the only critical point of the function, it is also the absolute minimum. Therefore, the function \(g(x)=5 x^{2}-20 \sqrt{x}\) has a relative and absolute minimum at \((1, -15)\).
Step 7 ：\(\boxed{\text{The function } g(x)=5 x^{2}-20 \sqrt{x} \text{ has a relative and absolute minimum at } (1, -15)}\)
