Graph question (1 Viewer)

[BlueGas]

I need someone to explain how to get the answer for all three parts please.

 

[kawaiipotato]

(i) When it asks for range, it's asking for "what values of y exist for this graph?". Looking at the graph, the y values can go all the way down to -infinity (shown by the arrow on the curve on the right hand side) and it can go up to y = 3 (stationary point. It doesn't go to +infinity because on the left side of the graph, it stops at y = 0. So the maximum y is y = 3 and minimum y is y = -infinity
Range: y is less than or equal to 3

(ii) When it asks for f'(x), it's talking about the gradient of the tangent on the curve. Imagine a straight line tangent at the starting point of the graph (-6,0). It would look like a straight line with a negative gradient. Going from left to right, the tangents slowly approach zero until x=-3 where it is zero.
After this point, the tangents start to have a positive gradient (ie. f'(x) > 0 ) until x = 2 where the gradient is zero and it starts to decrease after this point.
So f'(x)>0 when tangents to the curve are greater than zero.

(iii) For f''(x) you can think of concavity. (concave up parabolas or concave down). It wants f''(x) > 0 so it wants a "smiley face" (how i remembered it) ie. a concave up parabola. If you look at the graph and trace the graphs, you'll be able to draw a concave up parabola between x = -6 and x = 0
and a concave down parabola from x = 0 to x = infinity.
So f''(x) >0 means positive concavity which occurs in: -6 =< x < 0 (no equality at zero because there is a point of inflexion which is f''(x) = 0)

 

[calamebe]

(iii) For f''(x) you can think of concavity. (concave up parabolas or concave down). It wants f''(x) > 0 so it wants a "smiley face" (how i remembered it) ie. a concave up parabola. If you look at the graph and trace the graphs, you'll be able to draw a concave up parabola between x = -6 and x = 0
and a concave down parabola from x = 0 to x = infinity.
So f''(x) >0 means positive concavity which occurs in: -6 =< x < 0 (no equality at zero because there is a point of inflexion which is f''(x) = 0)

Would x=-6 be inclusive, though? The curve doesn't have a gradient or a value for f'(x) at that point, so it wouldn't have concavity, or f''(x) wouldn't exist at x=-6. Or do we just take the limit of the derivative as x approaches -6?

 

[kawaiipotato]

Would x=-6 be inclusive, though? The curve doesn't have a gradient or a value for f'(x) at that point, so it wouldn't have concavity, or f''(x) wouldn't exist at x=-6. Or do we just take the limit of the derivative as x approaches -6?

Hm, I think it would still be inclusive. It's continuous