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Q:

A curve with equation $$y=ax^{3} +bx$$  passes through the origin at an angle of  $$45°$$

Passes through the point $$( 2 , - 6 )$$

a) Find the values of $$a$$ and $$b$$.

Q:

Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval?

$$\text{f}(x)=5x^2-2x+1,~[0,~2]$$

a. Yes, it does not matter If $$\mathrm{f}$$ is continuous or differentiable, every function satisfies the Mean Value Theorem.

b. Yes, $$\mathrm{f}$$ is continuous on $$[0, 2]$$ and differentiable on $$(0, 2)$$ since polynomials are continuous and differentiable on $$\mathbb{R}$$.

c. No, $$\mathrm{f}$$ is not continuous on $$[0, 2]$$.

d. No, $$\mathrm{f}$$ is continuous on $$[0, 2]$$ but not differentiable on $$(0, 2)$$.

e. There is not enough information to verify if this function satisfies the Mean Value Theorem.

If it satisfies the hypotheses, find all numbers $$c$$ that satisfy the conclusion of the Mean Value Theorem.