the graph of the function has one relative extreme point. plot this point and check the concavity there…

the graph of the function has one relative extreme point. plot this point and check the concavity there. using only this information, sketch the graph f(x)=4x^2 - 5\nthe relative extreme point on the graph is (type an ordered pair.)

the graph of the function has one relative extreme point. plot this point and check the concavity there. using only this information, sketch the graph f(x)=4x^2 - 5\nthe relative extreme point on the graph is (type an ordered pair.)

Answer

Explanation:

Step1: Find the first - derivative

Differentiate $f(x)=4x^{2}-5$ with respect to $x$. Using the power rule $\frac{d}{dx}(ax^{n}) = nax^{n - 1}$, we have $f^\prime(x)=\frac{d}{dx}(4x^{2}-5)=8x$.

Step2: Find the critical points

Set $f^\prime(x) = 0$. So, $8x=0$, which gives $x = 0$.

Step3: Find the $y$ - coordinate of the critical point

Substitute $x = 0$ into the original function $f(x)$. Then $f(0)=4\times0^{2}-5=-5$.

Step4: Find the second - derivative

Differentiate $f^\prime(x)=8x$ with respect to $x$. We get $f^{\prime\prime}(x)=\frac{d}{dx}(8x)=8$. Since $f^{\prime\prime}(0)=8>0$, the function is concave up at $x = 0$.

Answer:

$(0, - 5)$