Question

Which of the following could be true if \(g^{\prime \prime}(t)=1 / t^{3 / 4} ?\) (a) $$g^{\prime}(t)=4 \sqrt[4]{t}-4 \quad \text { (b) } g^{\prime \prime \prime}(t)=-4 / \sqrt[4]{t}$$ (c) $$g(t)=t-7+(16 / 5) t^{5 / 4} \quad \text { (d) } g^{\prime}(t)=(1 / 4) t^{1 / 4}$$

Short answer

The options (a) and (c) could potentially be true if \(C1=-4\) and \(C2=-7\) respectively. Option (b) cannot be validated due to missing information, and (d) does not match our calculations.

Step-by-step solution

Find \(g'(t)\) from \(g''(t)\)

We start with \(g''(t)=1/t^{3/4}\) and we integrate it to \(g'(t)\). Using the power rule of integration: \(\int t^n dt = (1/(n+1))t^{n+1} +C\), we get \(g'(t) = (1/(1-(3/4)))*t^{1-(3/4)}+C1 = 4t^{1/4} + C1.\nSo, \(g'(t) = 4\sqrt[4]{t} + C1\).

Compare \(g'(t)\) with options

Looking at the options, we see that (a) \(g'(t)=4 \sqrt[4]{t}-4 \) and (d) \(g'(t)=(1 / 4) t^{1 / 4}\) are potential solutions for \(g'(t)\). The form without -4 in option (a) matches our calculation. Therefore, we can say (a) could be correct if \(C1=-4\). However, (d) can't be right as the coefficient of \( t^{1 / 4} \) does not match our calculation.

Find \(g(t)\) from \(g'(t)\)

We integrate \(g'(t) = 4t^{1/4} + C1\) to get \(g(t)\). For \(4t^{1/4}\), we have \((1/(1+(1/4)))*4t^{1+(1/4)} = t - (16/5)t^{5/4}+ C2\). Adding \(C1\) (a constant) will still give us a constant that we can denote as \(C2\). Therefore, \(g(t) = t - (16/5)t^{5/4}+ C2).\n

Compare \(g(t)\) with options

Looking at the options now, we see that, (c) \(g(t)=t-7+(16 / 5) t^{5 / 4} \) is the potential solution for \(g(t)\). The form matches our calculation. Therefore, we can say (c) could be correct if \(C2=-7\).

Conclude

From the options (a), (b), (c), (d), only (a) and (c) could be possible if the constants of integration are -4 and -7, respectively. These are determined by initial conditions which are not given in the problem. Therefore, we can't validate any other options based on the given information.