Definition of a probability density function:
If f(x) is a probability density function, then \(\displaystyle \int^6_0 f(x)\,dx = 1\).
This means for the probability density function in the question, \( \displaystyle \int^6_0 Ax(6 - x)\,dx = 1\).
Now, notice that \( \displaystyle \int^6_0 x(6- x)\,dx = 36\).
Therefore A = 1/36.
For part B, sketching a quadratic function should be easy.
For part C, just find \(\displaystyle \int^6_4 f(x)\,dx\).
For part D, let W be the weight of a random jar. We can find \(\mathbb E(W) = \displaystyle \int^6_0 xf(x)\,dx\) and \(\mathbb E(W^2) = \displaystyle \int^6_0 x^2f(x)\,dx\) and then use the formula \(\operatorname{Var}(W) = \mathbb E(W^2) - \left(E(W)\right)^2\)
You should know how the standard deviation and the variance are related.
