Answer to Question #268734 in Calculus for hello

Question

Determine the concavity , y-intercept , x-intercepts and co-ordinates of vertex of the parabola 𝑓(𝑥) = 5𝑥 2 − 𝑥 − 3. 

Answer 1

To find concavity we should find the second derivative

$f'(x)=10x-1$

$f''(x)=10>0\implies$f(x) is concave upward on the whole domain

To find y-inercepts we should put x = 0

$f(0)=5*0^2-0-3=-3$

point (0, -3) is y-intercept

To find x-intercepts we should put y = 0

$y=0\implies 5x^2-x-3=0$

$D=1+60=61\implies x_1={\frac{1+\sqrt{61}} {10}}, x_2={\frac{1-\sqrt{61}} {10}}$

points $({\frac{1+\sqrt{61}} {10}}, 0), ({\frac{1-\sqrt{61}} {10}}, 0)$ is x-intercepts

Let point (a, b) be a vertex, then

$a={\frac {-b} {2a}}={\frac 1 {10}}$

to find b we should find f(0.1)

$f(0.1)=5*0.1^2-0.1-3=-3.05$

point (0.1, -3.05) is a vertex of the parabola